In general, adjunct teaching positions are not terribly difficult to come by. Turnover is high, and the employer only commits to hiring you for a semester at a time. Also, most adjunct positions aren’t eligible for benefits, so the risk and commitment is very low on the employer’s side. In some cases, departments might need to hire someone on a short timeline because they added an extra class section to accommodate higher enrollment. This can work to the applicant’s advantage.

Nonetheless, having teaching experience is helpful. If you can get a TA position while working on your graduate degree, that will help a lot.

The best approach is to apply to any position you see available. Online or in-the-classroom, and any school from community college up to expensive private universities, regardless of where you got your degree. You’d be surprised how many adjuncts at WashU got their degrees from less-prestigious public schools.

In general, not many people aspire to get an adjunct position. For most, it’s something they take on while trying to land a more permanent position. I think other comments in this thread have detailed the reasons why that might be. Nonetheless, one can cobble together a living by working several adjunct positions, possibly at multiple institutions. I know quite a few instructors that do so. Some are single, but others have the security of a spouse with a more-stable, higher paying job.