You don't need a TEFL course to teach English in Asia, you just need a Uni degree. That's the minimum requirement.
If you have a Master's then you could potentially teach at a Uni.
What really matters is the country you want to go to. Seoul, South Korea has one of the best pickup communities but it's notoriously difficult there. Japan is a good midway, not nearly as difficult, really quality women, and nice lifestyle. But it's expensive and the community there isn't so great, so you'd often be picking up alone.
Shanghai and Chengdu, China are pretty good. Nice quality women, low expensese, good community. But you have to have a certain fortitude to handle mainland China. Taipei, Taiwan is ok if you want primarily one night stands, but like Japan, the community isn't that great for pickup.
So I'd boil it down to where you want to go. Weigh the pros and cons of each place, or you can also ask our opinion and we'll give you our feedback.
Japan and Seoul need the highest initial savings and the highest quality of life. Seoul needs more upfront money but isn't so expensive over the longterm while Japan is more expensive overall. China is the best option on a tight budget.
I didn't mention Southeast Asia which aren't bad choices either, if you care more about lifestyle and climate and want the women to be more receptive to your advances.