I am a dinosaur. I remember the times when a person could stay in Taiwan if they wanted to for up to six months , with two sixty day extensions at the local precinct as long as they had a job. And companies were not restricted from hiring foreigners.
That was definitely the last century. Today, Taiwan is a place where tons of young Taiwanese people have MBAs and other advanced degrees and are working for very modest salaries. So there is a huge pool of "acceptable" talent for most enterprises.
The island is one of the most crowded places on earth. Supporting some 23 million people on a rock 300 miles long by 100 miles at its widest with two thirds of that mountains.
Competitive? Yes, very. With laws making it very hard for companies to hire foreigners, the days of rocking in to Taiwan and just job hunting is not usually workable, EXCEPT for those intending (and qualified , with the "right" passport) to teach English. Basically all other professions, you would be hard pressed to D-Day over here and expect to be gainfully and legally employed in a NON English teaching job in any quick time frame.
The laws are geared against companies hiring foreign passport holders except companies with a lot of pain in capital and then only for people highly specialized and qualified. WE are talking White Collar here folks.
Given all that, perhaps a good course of action (unless one had a huge stash of cash to draw from where you could spend years just doing visa runs and being opportunist) would be to contact the big Taiwan companies in the field you want to work in. See if you can get solid interest and interview over the internet and line up a suitable job before you got to the rock to stay.
Of course, best to visit for at least a month and just rove around the cities and the rest of the rock and gain a deeper understanding of IF you really want to Lunar Module onto the surface. If you do, make sure you have enough juice to lift off back the heck out if things get unworkable. And they most likely will .
Taiwan is a niche place . It is no Singapore or Pre China HOng Kong.
It is suitable for the young college grad with the right passport to goof around Taiwan and Asia for a couple of years. And unless that said person plans to marry a local and just feels at home on the rock, it will eventually sour.
You really have to be at least half Taiwanese in fact or at least in spirit and MIND to go long term on the rock.
Other then that just get in and get out. Do what it is you want to do, learn what it is you want to learn and high tail it out of Dodge when mission accomplished.
Taiwan is like eVA Airways. They train the young foreign pilots, make em work and then the pilots boot on out to other countries and other airlines once they have the experience and "time" under their belt.
Taiwan can be a great LEARNING experience. But it's the dark side of the moon. Do you REALLY want to live there long term? Probably not, unless its in your BLOOD or you have been there long enough to be more Taiwanese then otherwise.
But if you have the funds and the inclination , by all means ADVENTURE away.