Most secured cards will let you set your own CL based on how much money you deposit into the account that secures it. MY personal choice would be, if you are looking for a secured card, I would go with a credit union. Either a local one to you or one of the larger national credit unions. Usually you will encounter lower/zero fees and they are just usually more flexible overall in my prior experiences. But that is just my personal preference, I've always had better relationships with CU's versus banks, but of course, YMMV.
Bank of America. I had a secured with $500 deposit, decided to put another $500 when the 4th statement cut (so hey you can add to it). I called to have it unsecured at 5 months and they returned my deposit plus gave me another $500 so now I'm up $1500. And 2 days ago after the 8th statement cut, I asked for $3000 and they gave it to me ( I'm mad, I should have asked for more). So had this card 8 months and took it from $500 to $3000. Can ask for a cli every 3 months.
Discover is making me call to add more deposit so that's not gonna happen. Earliest it can unsecure is 7 months and I hear they are stingy with CLIs so we will see after I get this unsecured hopefully soon.
Capital One: They have been a pain when it comes to CLIs. I regret them being my first secured card.
Personally, I'd recommend Discover's secured card. Not only can you change to the 5% rotating categories rewards program, but they unsecure as early as 7 months. There are also no fees and Cash Back Match can be quite valuable.
You *might* squeeze an unsecured Cap One Platinum card out of your existing profile. (How long ago was the deliquent payment(s)?) The Platinum can usually be PC'd after a few months into the no - AF Quicksilver, where you'll earn 1.5% cash back on all purchases.This will probably be a credit limit bucketed card, but it is one of the best unsecured builder card routes out there. Use it until your profile outgrows it, then you can either sock drawer it or close it. (Stay away from the Quicksilver One as you'll never shake the $39 annual fee. The route I explained with the platinum is to achieve the no annual fee Quicksilver.)
While Cap One does offer secured cards, they tend to end up bucketed into low credit lines, even after eventual graduation, where they'll languish until you choose to close them, even if your scores have grown into the 800's and you're carrying $30K+ CL's from Chase, Citi, USB and everyone else.
For secured, I'd definitely recommend using a credit union. Navy Federal is solid and well loved around here, but you need a military affiliation whether it be yourself, spouse, child, parent or grand parent.
Discover seems to be decent as well, but be aware that they seem to sporadically ask for 4506_T (tax info), even on secured cards!
US Bank (and it's Elan branch where subscribed financial institutions offer Elan backed card offerings), BoA and Citi offer secured cards which will graduate and can be PC'd into rewards products.
Best of luck and keep us posted!
