I would say that as a rule of thumb day hiking permits are free and can be applied for on the day of the hike if they are needed at all.
Some exceptions of course apply
Really popular hikes that have a daily quota to reduce impact like the Zhuliu Old Trail in Taroko - I think it is really cheap/free but it is a really popular hike and you need an advance permit. http://opositlife.blogspot.tw/2013/09/h ... trail.html
Another example is the Alanyi Coastal Trail in southern Taiwan that goes through aboriginal land. This is a very popular dayhike and they require you to hire an aboriginal guide (3000NT for a group of up to 20). http://theworldisnotthatbig.com/2015/05 ... il-taiwan/
Other areas might require you to get a police permit on the day of your hike. Typically these are not limited and is nothing more than filling out a form for a few minutes. Beidawushan is an example even if you only want to dayhike part of the trail.
I just read a blog about a couple that almost got turned back dayhiking the Walami Trail because they forgot their permit in their scooter. 5 years ago when I hiked the trail they didn't require this. I tried to get an overnight permit but they were overbooked already and they didn't issue a dayhike permit. There was a ranger in the parking lot checking license plate registration at 5pm when I returned.