Restricting ourselves to recent history, from the first televised concession speech given by Adlai Stevenson to President Eisenhower in 1952, a great source for this information is
Paul E Corcoran's article in the Political Communication journal entitled Presidential concession speeches: The rhetoric of defeat. It looks at the concession speeches from Stevenson until Bush & Perot in 1992. In particular, a surprising number have deviated from the predictable election night/morning private phone-call/telegram, public speech pattern.
Although Nixon made a public, televised speech on election night at around 3:00AM EST, he doesn't necessarily concede the election.
And I—as I look at the board here; while there are still some results
still to come in—if the present trend continues, Mr.—Senator Kennedy
will be the next President of the United States. I want all of you to
know, I want Senator Kennedy to know and I want all of you to know
[sic] that certainly if this trend does continue, and he does become
our next President, that he will have my wholehearted support and
He also hadn't contacted Kennedy privately beforehand to concede, instead saying that he "presumes that he probably is listening to this program". A telegram was released congratulating Kennedy the next morning. Corcoran also notes that a report of Kennedy's victory speech noted that "The Senator had stayed up until 3:50
a.m. awaiting [a] concession and had gone to bed disappointed when
the Vice President withheld it."
In a similar style to Clinton in 2016, Barry Goldwater didn't give a televised concession speech on election night, instead opting to wire President Johnson privately - he gave a news conference the following morning. In another parallel to 2016, Johnson gave his victory speech on election night despite the lack of a public statement from Goldwater.
Dole conceded the election at about 6:00 AM the morning after, according to the timestamp on C-SPAN's video of the speech - later than usual, but still arguably during election night. He also conceded to Clinton privately beforehand.
The most obvious example of a lack of concession speech on election night is Al Gore's concession to Bush which took place over a month after election night, after the Supreme Court issued its ruling on the Florida recount. According to Scott Farris' book Almost President: The Men Who Lost the Race But Changed the Nation;
Gore had said that on election night he felt considerable pressure “to
be gracious about this,” which led him to concede perhaps too quickly
when his own interests would have been served by waiting a while
longer. Gore had called Bush to concede at about 2:30 a.m. EST the
morning following Election Day, and that call was widely reported in
the media. It was while Gore was en route to give his formal public
concession speech that aides intercepted him and urged him to retract
his concession because of tightening vote totals in Florida. Gore
again called Bush who, incredulous, asked, “Let me make sure that I
understand: You’re calling back to retract that concession?” Gore
replied, “You don’t have to be snippy about it.”
The most recent example, Kerry didn't concede until the afternoon of the day after the election. This was due to his campaign's ability to remain in contention being reliant on uncounted ballots in Ohio. Once it became mathematically impossible for the Democrats to carry Ohio, he gave his concession speech, noting that he had conceded privately to President Bush in an earlier phone call.
Hillary Clinton conceded privately to President Trump on election night, at around 2:30 AM according to Kellyanne Conway, or just after AP called Pennsylvania for Trump at 1:35 AM according to Hillary in her book, What Happened. She then gave her concession speech the morning after.
In conclusion, then, although this is slightly out of the ordinary, it is not without precedent. While it lacks the same circumstances that delayed the concession speeches of Kerry & Gore, it is comparable to Dole's in 1996, and very similar to Goldwater's in 1964. A private phone call, or earlier, a telegram, to concede to one's opponent privately prior to a public speech has been made in every case since Nixon in 1960.