I was offered a second work permit last year, and had a bit of a rigmarole dealing with what the employers thought was the correct procedure. Eventually I decided it was too much trouble and quit the second job. The problem was not the actual procedure, it was all the different interpretations by employers.
As far as I could tell definitively, the second employer simply applies for a second work permit in the usual way, as if it was a ‘normal’ one, with no need to involve the first employer. ie you go ahead and get a medical, supply all your documents, etc. But don’t be surprised if you get asked for permission from your first employer, or some evidence that this second job won’t interfere with the first. (Might be difficult if you’re already employed full-time.)
You’re supposed to guarantee 14hrs a week of work for a ‘first’ work permit, but I don’t know if that holds true for a second WP. If it does there’s no reason why you can’t pay the teacher for prep time and insist on a ratio of prep:classroom that suits you. And you can agree what you like as a pay rate. eg 1hr prep to one hour of teaching, all paid at 500/hr, instead of 1hr teaching at 1000//hr and unpaid prep. You could also offer people Xhrs at $Y/hr writing course material or something, with X times Y coming to less than they will be actually earn.
When I was applying for ARCs for teachers last year I had to provide timetables of lessons for the whole school so that the authorities could see how many hours the teachers were scheduled to teach. I also had to provide a list of everyone we were employing, including admin staff etc. This was in Taipei City.
I believe that WPs come in 3 month, 1 yr,2yr, and 3 yr options. Is there a 6 month option too?
Once you get the second WP I don’t know what happens next. Do you need to get your re-entry permit extended? I believe ('cos I read it here somewhere and 'cos it makes sense) that you have to go to the police and get your ARC amended.