Welcome to the myFico forums!
Please check the sticky thread at the top of the forum, 'Helpful Rebuilding Threads' and inside of it is a link to a chart showing many of the secured cards that are available.
Some of them do not require a credit check, so if your deposit clears, you're in.
Also, make no new baddies on your credit and just paying on time will help, and the ageing of your current listings will help too.
Sorry, just reviewed the list and it doesnt readily indicate the ones that do not require a credit check.
From memory, I believe Open Sky is one that does not check.
You might also try the FingerH ut rebulding or starter program too.
eta: If you are Ok in the Chex Systems, you could also try SDFCU, Sate Department Federal CU online. The 'who can join link' will show you how to join. You can make a deposit of like 250 and then ask for a secured card.
OpenSky definitely does not pull credit, and it reports quickly to all 3 bureaus. Requires a minimum of $200 to open and has a $29 annual fee. Good luck on your journey.