In my experience working for banks we were required to run Chex even when opening savings accounts, however, it was easier to make an exception to open those types of accounts because they weren't as easily accessible (no debit card or check writing capabilities).
If you're running into issues opening new accounts because of chex try going into a local bank or credit union and explain in advance what your situation is. Most bank managers have the ability to override a Chex decision unless the reason is due to trying to defraud the bank. Tell your story and let them know that you're working to turn over a new leaf. It will really help if you have a letter from the bank who initially reported you stating that you're working to rectify the sitatuion (via a payment plan or otherwise).
I agree with the other posters that going with a finance company is a bad idea because of the very high interest rates and fees. At the bank I worked for we did not put much weight on Chex status when decisioning loans unless it shows up as a collection on your credit report.
Hopefully this helps!
My Scores as of May 20, 2015
Experian - 757
Trans - 769
Equifax - 754
