For UA/MP-issued Star Alliance award travel one can change the itinerary prior to flying the first segment. There may be fees/charges associated with changes. (As a 1K I do not pay any fees/charges.) Once travel begins you cannot change the itinerary. One can envision all sorts of changes, outside the rules, but these are irregular at best. (I did change UA/MP-issued award travel on LH with LH after the trip had begun; I am a SEN with LH and believe this helped my situation. I changed the day, flying one day earlier and LH changed the routing.
Can you share your itinerary in a bit more detail? And/or explain what it is you are trying to do? Have you started your trip yet?
If you haven't started your trip yet then just ring UA/MP frequent;ly to inquire about "I" inventory on TG on the new date. If travel has begun you could approach TG about a change but I suspect they will refer you back to UA. You could also try to stand-by at BKK but I can see TG getting confused and not allowing this option.
BTW, I was able to get BKK-JFK on TG in J using UA/MP miles quite easily, the exact date I wanted, in January for May travel.