I had this whole thing written, but re-reading OP's post I see he's interested in a marketing position. I have no idea about the marketing field. So I guess this is more a general job info post. For teaching, the best time to apply is from now until about May or June when schools start making hiring decisions. Some places have rolling deadlines. Teaching doesn't pay great (certainly nowhere near the salary you want for that marketing manager position), but I find it rewarding.
IMO, your ability to get a good job depends on three criterion:
(a) Your level of experience and qualifications, (b) Where you are, and (c) Who you know.
For example, I have a full-time university job in Kaohsiung. I have a Masters and almost a decade of ESL teaching experience, but I don't think I could have gotten such a sweet gig in Taipei unless I had a PhD as well. My wife is also a teacher and one of her old professors is part of the faculty and recommended the position. So there's a variety of factors that come into play. Taipei is difficult these days. Unless you're cool with an average cram-school position, I'd suggest checking out Kaohsiung, Taichung, Tainan or some other place. I assume if teaching jobs are easier in the smaller cities, then other jobs such as the one OP desires may be as well. Speaking Mandarin is a plus and will certainly help you.