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I would wait till your home loan reports on your CR and seed what happens to your scores. Also if your mortgage is with a local CU, they could offer your a credit card along with the loan.
First, you need to make your EX great again. Then apply for Amex because Amex mostly use EX. Use the Amex travel cards and/or Hilton cards.
I would go with NFCU for a new card AND a CLI on the card you currently have. Dont know who you bank with but, all my banking is with them and it is very convenient and easy to manage everything through their online banking site. Customer service cant be beat and they tend to be very generous with their SLs and CLIs.