Trinket Wars is a great gamemode, mostly because it's Deathmatch with a twist, I just love deathmatch.
However, it has so many fatal flaws...
Whatever it is, whatever the reason, here are my suggestions;
1. EXTREMELY DESIRED FEATURE: Make it so when the Trinket Carrier can 'lock-on' to a player, an icon will appear on top of his head, meaning he's able to directly "Pass" the trinket.
When he drops the trinket, it is automatically passed AND ACQUIRED/OBTAINED by the 'locked-on' player.
2. Add an ability to give "throw strength", so whenever you hold the 'dropitem' button (default G), you'll see a power bar. If you just tap G, you drop it at around 10 force, essentially like you're giving it to another teammate, not just blatantly throw it like a monkey trying to throw like a human (humans got the best throw strength of all animals).
(As for when you die, just make the throw strength be the same as in vanilla, which should be the max throw force anyway)
3. Add a warning that you can't drop the trinket because you've just got hurt, most people don't notice that even some semi-veterans get confused about it.
Should work as a hint, hence if hints are disabled, it won't appear.
4. Add an extreme in-your-face WARNING that you can drop the trinket by pressing G. Also, perhaps some way for whenever somebody says "Press G" or "Drop the trinket" the player will get a warning that details how to drop it in his native language?
5. Make it so even if the trinket is dropped you spawn near it, or at least somewhat-near it. It's really unfair, in tw_temple, how you can sometimes spawn far away from the Trinket CARRIER, meanwhile, if the Trinket is dropped, the player is way too far to reach it even in the very last second, even as an Enraged Berserker or Skirmisher.
6. Add a better HUD Indicator that you have a trinket, not just that heavily distracting "Team-based Color Correction", such as make it so when you pick the trinket, you get an in-your-face overlay that then turns into a small icon on the bottom or so. Kind of like "Spy Cloak" in Team Fortress 2.
(I'm willing to make an icon of it myself if anyone else is willing to make the script/code for that.)
7. Make it so when the trinket is not carried, there's not just a timer but also an indicator on top of the trinket that clearly states you have to press [E] (or any key binded, if possible) to acquire the trinket.
8. Just like the indicator, there should be a proper timer (right in the HUD itself, not outside acting as a World UI) that you can toggle on or off in the options which should show when the trinket will disappear AND when will the trinket appear if it did disappear.
Not just that, should also be able to make a small top-head indicator showing who is to get the trinket.
(These next ones are 'different' simply because I bet they will make absurd arguments and debates, not just that, it's more a job for "somebody else" than a developer)
9. Add an ability to vote for somebody in the team to drop the trinket, fancily call it "Mutiny Vote". To avoid any arrogance, maybe make it so when the Trinket carrier gets more than 2 kills himself (including assist) he's immune to it until a minute has passed.
10. Add an ability for admins to force a player to drop the trinket. Could be as "complex" as pressing a button which, when you aim at the target, it drops it.
11. If 4 and 5 is heavily undesired, why not make it so the Trinket is given randomly, but on the next round, the higher scores get a better chance of getting it than others?
12. Add an option to remove the particularly badly made Color Correction used for when you are carrying the trinket.