Been a Long time since I had trim fall off, and it was on my Z show car. What I did was pull it off CAREFULLY....remove ALL the adhesive on the trim piece then clean that and the area whre it goes with alcohol, I use 90%.
3M makes a double sided trim tape, and they have it at Wally mart, you can buy it in different widths. Put that on the trim piece then remove the door side paper and press in place from right to left NOT JUST STICK THE WHOLE THING ON from one end to the other slowly. Once done your good to go and if you cleaned the trim and door well enough it will not come off again!
The 3M tape is black with a red adhesive protector on both sides, it should be around 5 bucks a pack. Buy the size appropriate for the trim not too thin or wide.
Or if you don't want to pull the whole thing off you can peel it away and use the same 3M product on that small section, BUT be prepared for the rest to fall away from the door eventually. I would just do the whole thing, ONCE AND DONE!
OH.... If you leave the piece out in the sun to heat up while you clean the door it will make it more pliable and thus easier to apply. Have the 3M tape on the trim piece while its heating..