Welcome to My Fico forums, @Anonymous! 
Great advice from @Anonymous above. I would echo the local credit union (or where you normally do your primary banking), Navy Federal CU (more than PenFed which can be more conservative), and Discover all are great targets for a next card.
You don't need to be a veteran or active military to get in with Navy Federal but if you're not, you will need a family relationship to a veteran. However, this includes spouse, parents, grandparents, siblings, or children so it's easy for many to find a way to apply.
In addition to the above, Capital One is a much better option at this point than going with the larger banks like Chase, Bank of America, or Citibank. If you don't qualify for an unsecured account, you may be offered a secured card from Navy Federal, Discover, or Capital One, or be able to talk them into via a "recon" (reconsideration) phone call if denied.
I decided to close my store-branded cards a few years ago, but for someone like yourself who is trying to establish credit, they can be a fantastic tool. Just don't go overboard with store cards. Limit them and try to pick places where you already shop and will find value from special offers or special financing. Synchrony and Comenity Banks underwrite a lot of these cards and they tend to be a little easier to get than some major bank cards.
There are a few deeper "subprime lenders" but if you can avoid them, I would. (Credit One is often mentioned here but there are others. While Discover and Capital One have some secured cards and partially cater to subprime, I don't consider it their primary market focus compared to lenders like Credit One.) However, if you get turned down by the above, they can be a viable alternative. The main problem with the Subprime lenders is that they can have high fees including annual fees and high APRs. Since they aren't long-term "keeper cards," you'll probably look at closing them eventually, and that's a negative factor when you're looking at closing one of your oldest cards.
You've gotten a ton of great advice here already OP. However, if it's not a bother could you give us a bit more detail about your profile? It might help make some advice more relevant than others.
- How many hard inquiries do you currently have?
- What is your average age of accounts?
- How many CC's and loans do you have?
- You mention you have an auto loan with great history, are there others? If so, how recent?
Like someone above mentioned, they don't approve based on scores. I've seen people with 680 scores approved for something that someone with a 730 score was denied for based on other factors like what I mentioned above. The more info you provide, the better the advice can be! Good luck OP!
Must garden until 2/1/2022 to hit my goal AAOA. 
Oh how simple this would be if the score was the only or at least main factor! And, I don't know why the score even exists if it's not the main factor, but, that's a topic for another sub-forum. As others have mentioned, your score is one factor among many. They may require a minimum score, but if so, it would be mainly used for weed-out. More info about your profile would be helpful in offering suggestions.
But I will add Amex to the list of lenders to check, especially, Amex cobrand cards like Delta and Hilton. I would go right now to the Discover site and check prequal, then go to the Amex site and check prequal. Do not pay attention to "prequalification" from other sites such as Credit Karma, Experian, Cardmatch, etc, they are not real. The ones on the Discover and Amex sites are real, and pretty accurate. They are not a guarantee. Not only will that potentially help you get a card from one of those two lenders, it will give you another view into how credit card companies in general view your profile.
Good luck!
