So this is a question asked out of laziness.
My employer pays for me to commute once a week for a meeting (let’s say Taipei to Taizhong), but in the morning I want to stop farther on (say Changhua). Can I buy two tickets for the same train? (Taipei–>Taizhong + Taizhong–>Changhua). How does that affect the putting your little paper ticket through? In theory I could get off in Taizhong, buy another ticket and then get on the next train, but obviously I’d prefer to make this faster.