Some questions to consider:

• Is price a factor?
• Do you have/need a visa for China?

The most interesting destinations: I'd say Kinabalu (BKI), HCMC/Saigon (SGN), Hanoi (HAN), also Bangkok (BKK, DMK) and Chiang Mai (CNX) but these two are more like 3.5h. Singapore (SIN) has lots of flights but is ~ 4h.

Seoul-Gimpo (GMP) from Taipei-Songshan (TSA), no need to go Taoyuan, saves the ~ 1h commute. Also lots of China destinations from there, as well as Tokyo-Haneda (HND) but not really cheap.

Also Osaka-Kansai (KIX), Okinawa-Naha (OKA), lots of other places in Japan but most only on particular days of the week, Macau (MFM), Manila (MNL). There is a direct flight to Puerto Princesa (PPS) if you plan in advance.

Hong Kong (HKG) is always TWD ~5k return. Not the cheapest to stay though, if price is a a factor, and honestly not that interesting either these days.

If it's not supposed to be a visa run, consider Taiwan's outlying islands: the most interesting are Matsu (MFK) and Kinmen (KNH).

Something more exotic? Try Palau (ROR), a direct flight on CI.