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True, mine bumped up around 20 points (?) when I got my first card. The thing to remember is, don't USE the card for anything big until after you've got the mortgage - just small purchases, and pay in full before you even get your statement. The mortgage co wants to see new credit, not new debt!
Good luck!
I'd try Orchard bank and Credit One. They are both rebuilders. Stay away from First Premeir if you can avoid it as their fees are very high. With your scores you will probably have luck with the two I mentioned. Plus, they have a preapproval so you have a pretty good chance of where you stand with them before taking the HP.
Didnt get approved for the no hassle card but we did go for the secured one. thru capital one
Orchard is owed by HSBC and will soon (very soon) become Cap One, Credit One is Credit One Financial which is a wholly owned subsidiary of Sherman Financial Group, LLC, the same company that owns LVNV Collection Agency. LVNV seems to play by their own rules (see link below).
Orchard perhaps, although I'd avoid them because of the high AF and low credit lines, but Credit One ? Never, ever, ever...
If the OP is unable to get an unsecured CC from a fair lender, then I'd go with a secured CC from a major lender like his/her bank, CU or Cap One (depends on the terms).