Again, if your redemption ticket is not with EVA and you have yet to commence your trip, you need to contact the airline that booked your award for re-booking. EVA will not touch those tickets.

If your non-EVA redemption ticket is half way used, you might have a case as EVA can control those tickets. In the past EVA has been neglecting these but there are successful cases of getting the ticket changed. YMMV.

If your non-EVA redemption ticket is partially on EVA, you have the options to try to get it changed beforehand, or fly until you reach the EVA segment and make EVA find alternative plans for you (that's if you are not stopped from checking in on the previous segments). EVA will have to sort it out as they cannot leave you stranded there mid-way. However, you do risk having to stay there till they find you an available seat.

I do not suggest you try to re-book these non-EVA tickets at TPE unless you have absolutely no choice but reached the D-Day. If you do have to go to TPE, be there as early as you can. You may be waiting for hours in line and receive an "I cannot help you." Try a local office (but many are closed today and/or tomorrow) if you cannot reach a live person on the phone.

Also, the tendency of EVA cancelling the late night departure to LAX and SFO has been a norm. You don't want to be there at midnight.