Welcome to My Fico Forums, @Anonymous.
Wow, looks like you burned most of the "big boys" in your BK and agree with FinStar that they are off the table for a very long time. 
I wouldn't even waste my time with any "IIB" lender right now, including Chase.
The Synchrony Amazon card is a great suggestion. Retail/store cards in general would be one more way to add new revolving accounts. However, make sure to add useful ones that you will naturally use and to keep them as a minor part of your overall lineup.
While rewards are great when you can get them, the more important thing for you to focus on now is that goal of rebuilding your profile. It's fine to have multiple cards with one lender and two cards with one bank helps your FICO just as well as-if they were with different banks. Remember that lender diversity is a smart goal, and putting too many eggs into one credit basket can be very risky if adverse action occurs. Just read recent threads on the forums of many members who lost huge credit limits with Synchrony as well as some other lenders, and some of those members had not diversified across other banks. Ouch. That can really hurt.
The largest national banks in terms of assets that you DIDN'T burn would be Bank of America and US Bank. You're probably not ready to apply for either of those now, but as you build your profile those would be the safest large national banks to go to first. Also, Truist is a new bank that is a merger of BB&T and Suntrust, and will probably be a large player going forward.
I would start with that additional Capital One card. Over time, I would add some good cards from credit unions. If you are eligible for membership (either by direct military service or through service of a family member including parents, grandparents, or siblings,) Navy Federal CU might be a good recommendation. Or just find a good credit union local to you, open some deposit accounts, and apply for credit.
Since you burned most of the largest national banks, another option would be to try to get a foot-in-the-door with more regional banks. Which ones would depend on your location. Some names that issue credit cards might include PNC, Regions, BBVA, Key, TD, Fifth-Third, BMO Harris, Union, M&T, Bank of the West, Huntington National, UMB, or FNBO. In some cases, the card products might not be as attractive as some of the larger lenders but building your credit is most important.
While installment debt is factored differently than revolving credit card debt, it will help your profile if you have an installment loan such as an auto loan in good status or repaid.
There is a "rebuilder" forum which offers much better support and information than the credit card forum. This is a good resource for you. Go to: https://ficoforums.myfico.com/t5/Rebuilding-Your-Credit/bd-p/rebuildingcredit
Good luck with the process!
