Do you have a specific goal, or just a general goal of higher scores at some point in the future?
Opening more cards now will temporarily drop your score but may have some benefit in the long run. Not opening more cards now will preserve your score in the short term but you may be more likely to get thin-credit denials in the future. If you are thinking long-term, I would probably try prequalification for some more cards I would keep longterm, and, I would probably get an installment loan of some sort; Self Lender is an easy one to get, but you probably qualify for other options. Having three cards gives a score boost and allows use of the All Zero Except One method. An installment loan will give you a "good credit mix" that gives a boost. But there are tradeoffs to everything. If you did that, you risk being shut out of Chase cards for a while (Chase has a notorious "5/24" rule that if you've opened five cards in the last 24 months, you are denied for any Chase card), and, you would get dinged for "credit-seeking behavior" and your average age of account would go down. So, in the short term, this would probably hurt.
I am not sure that a secured card with a high credit limit will do that much for you. Most (not all) secured cards report as secured cards, meaning, lenders can tell the difference between secured and unsecured. They count the same for the credit score but may be counted differently by human loan officers. Again, is there a particular reason you want a high credit limit (like do you need it for travel), or is it just part of a plan to have higher scores down the road? If the former, it may make sense for you, but if the latter, I would just wait to get higher unsecured limits.
There is no credit benefit to paying in full, in and of itself. What helps credit is paying on time, and keeping utilization below 9% at the time your statement is cut. Paying in full is a good practice to avoid paying interest and to ensure utilization doesn't creep up on you, but you don't get rewarded for it in terms of credit score beyond the factors I just mentioned.
Good luck!
If I were you, I would try Amex prequalification and see if anything pops up. If it does, I'd probably apply for it. Wait one to three months depending how conservative you want to be, check Discover prequalification. If you are prequalified for it, I'd get that. Then sit tight as far as cards. You will be close to setting off "credit-seeking behavior" alarms and your credit score may possibly go down due to age of accounts tanking, but then again, it may go up due to having three cards. If you don't get or don't want one of those two, another thing to consider is the Amazon Store Card if you shop a lot with Amazon. It's easy to qualify for and limits can build up there. I would probably also do a Self Lender loan. You don't actually get money you can spend. You "borrow" $500, they put it in a CD for you, you "repay" the "loan" over time, and at the end, you get the money from the CD. Since they charge a higher interest rate than the CD earns, it will end up costing you something like $60 or $70 altogether, but, you get a positive installment loan on your reports, which triggers the "good credit mix" FICO reason.
That's kind of a medium-aggressive plan I'm suggesting, but, I can't say it's the best plan, just that it's probably what I would do in your shoes. Good luck!