Ok, with my experience in hitchhiking through Taiwan, I have noticed a dead give-away that someone wants you to teach them or their children English. It’s the question “How long will you stay in Taiwan?”. So, my advice to you…if you don’t want to spend the remainder of your ride lying about how you’ll be busy on Saturdays, and how “Oh, Sundays are bad for me too”, or “You know, your house is very far away”…always say you’re only staying for a few more days.
On the other hand, though, if you’re looking for some extra work, go out on the road and stick your thumb out. I’ve almost always been offered a private-tutoring job.