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I would wait a minimum of six months, but it would be better to wait a year. That's a lot of new accounts in three weeks, especially on a thin file. You need to let the accounts age, show responsible use, and bring up your average age of accounts. Barclay especially doesn't like a lot of recent HPs or new accounts. If you want approvals for the HPs you'll take, I'd wait a year before applying again. Come join us in the garden!
I would personally avoid store cards unless you're really loyal to place and having the card results in significant gains.
Also i don't know your spending ability or habits but having too many cards and spreading the rewards all over the place for sometimes not even a penny on the dollar more doesn't really worth the hassle imo.
You have some great cards so far. And if you want to maximize your rewards and needs cards like the amazon card or premium cards down the road you need some strategy and patience not mindless app sprees. I'd say a 6 month period of gardening is warranted at this point. And figure out what you need and plan to apply just for it unless you want to go chasing subs and you're not really interested in the value of the cards.