Can a bill that was vetoed by one POTUS be reintroduced in its previous form and sent to the new POTUS? Keeping in mind that we also have a new Congress?
The same Congress could even send the same bill again to the same President (to be vetoed again) if they wanted to. Nothing stops Congress from passing anything allowed under the constitution as many times as they want over as many sessions as they want except themselves.
Generally it isn't ideal because things change. Bills are negotiated so when the negotiators change, or public opinions change, what was a reasonably balanced bill now isn't. Say the election shows more support for puppies than previously and less support for kitties. The puppy subsidy would be increased at the expense of the kitty subsidy if it was renegotiated so it may be viewed as unfair by the puppy supporters in the new Congress.
But more politics.
With a strong leadership and a well organized party it may be much easier to get acceptance of a sub-optimal balance without further negotiation than fight for whatever change in balance happened since the last negotiation. If the old bill is still favorable to leadership (who maybe are more interested in the pony spending anyway) they may be able to convince the puppy supporters to go along with the old bill to avoid all the hassle of renegotiating. They may offer promises of better benefits in the next negotiation because negotiations of non-trivial bills are a lot of hassle.