The turn player has priority to activate Exiled's effect and tribute itself before it can be
Trap Holed.
Nowadays the "proper" way to do things as the turn player would be to Normal Summon Exiled and immediately declare that you are using your priority to activate Exiled's effect.
Similarly, if you are not the turn player and your opponent simply summons
Exiled Force and says nothing, you would ask "Are you going to activate Exiled's effect?" If they say "No, not right now" they have effectively passed their priority, in which you can now respond with
Trap Hole and the turn player has lost the opportunity to use Exiled's effect, since they cannot chain to
Trap Hole.
This does seem a little tedious, but the priority issue has been a big issue for a while now and it's always under some form of confusion.
Remember. PLAYERS get priority, not cards.