Can a legal fair-yes or no-vote by a union membership be later be
rejected, because more then fifty percent of the membership felt they
voted wrong the first time?  The vote would probably swing the other
way the next time.

Most organizations with a voting membership have in their bylaws a
clause that allows a defined number or percentage of the members to
call for an extraordinary meeting for the specific purpose of voting
on a motion stated in this call for the meeting.
If the union had this type of clause, it would seem that the original
motion passed could be presented for a re-vote.  It then would depend
on the requirements for calling an extra-ordinary meeting  - which may
not be easy: signatures of 10%, 20%, ?% of the membership.  And there
would be a lapse in time, since there would be a minimum notice period
before the meeting could be held.
This would be significant, if action had already been taken after the first vote.