Amex Zync would be a good choice! Maybe Wal Mart Discover....
For practicality purposes, I will try Barclay first. The Apple Card seems to get the more approvals than any of their other cards.
The reason I will try that first is that you are more likely to get approved and you are more likely to get a decent CL. Compared to all your other options. Be prepared to call and recon. You will either be instantly declined or you will be given a less than ideal credit limit. Just call and they'll almost invariably approve you.
After about 3-6 months with the Barclay, you can then apply for other cards. True, Barclay's is inquiry sensitive but you will be okay as long as you don't go on an app spree after getting their card.
Zync is an AMEX charge card, payment in full is due about 2 weeks after the statement cuts. they want 6 months credit history on one or more cards, and a 620 or better fico they pull EX usually. Since you qualified for a home loan your score is most likely good enough.
For anything else your file is very thin, having your loan report may help things, you may want to try for a cap1 card now. Check there pre-approvals. 3 hard pulls... easy to get, but at a price 3 HP's and a annual fee usually.
After you have 12 months experience with 2 or more credit cards you should be good for Chase freedom, or something from citibank.
Homedepot (citibank co-braned easier than a full citibank card) shoud be possible very good for CLI's when you need them
If you can get your scores over 700 by the 12 month mark you chould then apply for a AMEX revolver like AMEX Blue Cash Everyday.
You could try for a Walmart Store (ge money bank) card currently, I think they like a 640 or better especially with thin files, you most likely you. It comes with your TU fico score monthly if you go paperless. You can get upto $60 per day over on purchases with no fees. CLI's every 4 months if you ask. All GE backed cards pull TU, and CLI are SP's
If you get Walmart Store card you could then apply for Amazon store (ge money bank) card good for no interest deals at $150 or above. same as the walmart card
Barclay's Apple can be used anywhere. It is really just a regular VISA card. It won't be an ideal card if you carry a balance- interest rate is super high but it has worked really well for me in terms of growing with me. I also opened two accounts after my Apple card and didn't get any AA from Barclay.
Since you are going to apply after you get funded, you can see if you qualify for the Zync since it will be a good card for backdating purposes. Then you can move on and try other issuers if you get denied.