I think the notion that Labour supports Brexit needs to be nuanced a little.
There's a May 2018 opinion piece in the Guardian (by prof. Anand Menon) that describes Labour's support for Brexit as "strategic ambiguity" because while supporting the result of the referendum, the LP doesn't really have its own clear plan for how Brexit should happen.
As you noted (and nicely detailed in Menon's analysis, which is agreement with another one from JRF, Labour's electoral gains (in seats) depend on constituencies that are pro-Brexit.
Menon's analysis also links to one of his colleagues who labelled the LP strategy "Brexit Blairism" and explained:
Labour’s decision to embrace departure from the EU in some form may have helped them reframe the election around other issues such as austerity and public services, and remind voters in Leave areas of their traditional suspicions about the Conservatives. Meanwhile in Remain areas, the party could advance by promising a “softer” alternative approach to “hard” Brexit.
The latter analysis does not have much in the way of concrete examples... but looking at Corbyn's Brexit speech, easily finds some, e.g. on immigration:
Our immigration system will change and freedom of movement will as a statement of fact end when we leave the European Union.
But we have also said that in trade negotiations our priorities are growth, jobs and people’s living standards. We make no apologies for putting those aims before bogus immigration targets.
Labour would design our immigration policy around the needs of the economy based on fair rules and the reasonable management of migration.
We would not do what this government is doing, start from rigid red lines on immigration and then work out what that means for the economy afterwards.
As Diane Abbott, our Shadow Home Secretary, set out last week, “We do not begin with, ‘how do we reduce immigration?’, and to hell with the consequences. Those are Tory policies and Tory values”.
To stop employers being able to import cheap agency labour to undercut existing pay and conditions, collective agreements and sectoral bargaining must become the norm. Labour stands for ‘the rate for the job’, not ‘a race to the bottom’.
But let’s also be crystal clear it is not migrants that drive down wages, it is bad employers that cut pay and bad governments that allow workers to be divided and undermined, and want unions to be weak and passive.
The last two paragraphs in particular echo the JRF findings that low-income, pro-Brexit/anti-immigration voters also want broader left-wing economic policies, whereas the paragraphs above those are an example of "softer" Brexit being proposed (on immigration in this case).
I tried to find some poll relating to the above. Alas, the most recent I found was from December 2017, but it does seem to support the "ambiguity" conclusion... around that time:
[A] recent poll by YouGov suggests that even people who plan to vote Labour at the next election are unsure of the party’s position.
Nearly a quarter said they thought Labour was completely against Brexit, a third thought the party was on the fence, and a tenth said they didn’t know.
In June 2018 there was actually a Labour "new single market" proposal that (methinks) partly contradicts the ambiguity claim on grounds of not proposing something concrete. On the other hand the June events confirm the "softer Brexit" view of Labour. As BusinessInsider [BI] reported:
Last week, the Labour leadership tabled an amendment to the Withdrawal Bill which detailed its policy of a new single market relationship with the EU, based on "full access" and "no new impediments to trade."
"We are confident we can build a new relationship with the EU. We want the UK to have a better deal than the Norway model," Jeremy Corbyn said to the House of Commons last week.
Labour's Brexit team believes its willingness to accept all EU regulations and standards would persuade Brussels to be flexible on issues such as free movement.
"In France and Germany, they are talking about the free movement of people. Standards are much more important to the EU than immigration," an ally of Shadow Brexit Secretary, Sir Keir Starmer, told BI last month.
Reactions from Brussels however were that the proposal was "cake-ism" and "selling a unicorn to paste over their internal divisions" (just like the Tories) as BI reported.
And if the question is mainly about Corbyn's position, back in February he supported a customs union after Brexit, unlike the government's view (then and still now) that that's too much.
Speaking in Coventry, a city which voted by 56 per cent to Leave, Mr Corbyn said: “We have long argued that a customs union is a viable option for the final deal.
“Labour would seek to negotiate a new comprehensive UK-EU customs union to ensure that there are no tariffs with Europe and to help avoid any need whatsoever for a hard border in Northern Ireland.”
“We are also clear that the option of a new UK customs union with the EU would need to ensure the UK has a say in future trade deals,” he said.
“A new customs arrangement would depend on Britain being able to negotiate agreement of new trade deals in our national interest.”
And Corbyn reiterated on that occasion that a custom union doesn't entail accepting EU migrants (obviously true, and happens elsewhere, e.g. Turkey has a customs union with EU). On the other hand, the reaction from the Tories was...
East Yorkshire MP David Davis slammed the Labour announcement, saying it would prevent the UK from realising the benefits of Brexit. The Tory Government proposes leaving both the customs union and single market.
The Tory position on that has eroded as well since February, but still not as much as to accept a customs union (and call it such).
And speaking of erosion, the harder line on Brexit has certainly sufferend that in June, with the rebellion of a significant number of Labor MPs:
Jeremy Corbyn has suffered a 90-strong rebellion over a Brexit vote on remaining in the European Economic Area, with six of his MPs resigning from their frontbench roles.
Their resignations were revealed moments before the result of a vote on a Lords Brexit bill amendment which called for the government to make remaining in the EEA a negotiating objective.
The House of Commons voted 327 to 126 to reject the proposed amendment, with 74 Labour MPs rebelling against their party's whip to vote in favour of EEA membership.
A further 15 Labour MPs rebelled in breach of the abstain order to vote against EEA membership, while one Labour MP, Susan Elan Jones, also defied the party's official position to act as a teller for the vote.
Labour leader Jeremy Corbyn said [...]: "I understand the difficulties MPs representing constituencies which voted strongly for Leave or Remain have on the EEA amendment to the EU Withdrawal Bill. "The Labour Party respects the outcome of the EU referendum and does not support the EEA or 'Norway model' as it is not the right for option for Britain.
"It would leave us with next to no say over rules we have to follow, it does not allow us to negotiate a new comprehensive UK-EU customs union and it fails to resolve the Irish border issue.
Even Reuters which usually doesn't advance much politcal commentary said of the June events:
But it was in the Labour Party where the deepest rifts were exposed. Many of its pro-EU lawmakers went against their leader, Jeremy Corbyn, by supporting the vote [on EEA] and not his amendment which argued for a new single market deal with the EU.
And if Corbyn wasn't too explcit on the meanig of this "new single market", his shadow Brexit secretary was:
Labour's shadow Brexit secretary Sir Keir Starmer said: "Labour will only accept a Brexit deal that delivers the benefits of the single market and protects jobs and living standards.
"Unlike the Tories, Labour will not sacrifice jobs and the economy in the pursuit of a reckless and extreme interpretation of the referendum result.
"Existing single market agreements that the EU has negotiated with third countries, including Norway, are bespoke deals negotiated with the EU to serve the best interests of those countries.
"We need to learn from them and negotiate our own more ambitious agreement, which serves our economic interests and which prevents a hard border in Northern Ireland. [...]
"Labour's amendment, along with a commitment to negotiate a new comprehensive customs union with the EU, is a strong and balanced package that would retain the benefits of the Single Market. Parliament should have the opportunity to debate and vote on it."
The "new single market" amendment was however defeated 322 to 240. Also on this occasion, the "customs amendment", which asked only for customs union after Brexit, had broader Labour support, being defeated only with 326 against 298.