Let sleeping dogs lie.
I think I will have to say, don't tell the old company about the unused ticket voluntarily on your last day of work, let them say the first word (i.e. let them ask). Otherwise you will open up a can of worms and you don't want that in the hectic time between ending one job and starting another.
Don't get the new company involved, this is also a can of worms, violation of the KISS principle, lots of back and forth, confusion, telephone tag, etc.
If the old company does ask, just say you will send the ticket itself back to them if they wish; don't elaborate and don't send them money instead.
Avoid buying tickets for business trips yourself, let the company book them. This way, if dates change, the company can and should arrange for modifying the tickets. This strategy won't work if you want to take your spouse and/or family and/or a vacationing co-worker along for their pleasure while you are on business.
Since your old company is a non-profit, if you want to give them back the value, wait a few months and make a tax deductible gift.
Travel tips:
http://members.aol.com/ajaynejr/travel.htm
[This message has been edited by AllanJ (edited 08-04-2002).]