Chase is not the easiest issuer to get a card from, especially with your lack of history.
It will be a few years before you can get into the Chase ecosystem, IMO based on lots of research. You can rush it and suffer the damage, but that will take longer to recover from than employing a smart, balanced approach. Having little to no history is many times worse than a tarnished history or even bad. With no history the lender doesn’t know what they are getting into. With a bad history, they have a pretty good idea of the risk. Some take that risk and extend credit at a ridiculous rate, others deny the application.
I would advise to learn how how scores are calculated, and what makes for a strong overall report.
Keep doing good with your little secured card, and let it mature. After you’ve had it for 12-18 months and it has been unsecured, look for an unsecured starter card. These are going to be entry level cards, most likely a basic cash back card. Use it wisely, always pay in full, and never miss a payment. Don’t let anything negative hit your report. $30, $250, $5000... nothing.
It may seem like a long row to hoe, but if you want to get into the top of the line cards then you have to put in the effort. Don’t fill out a bunch of applications, open easy store cards, etc. You need to build history, but you need to be smart about doing so. With your history it is very, very easy to muck it up with too many junk accounts, inquiries, bogus loans, etc. As tempting as it may be, stay the course. Your 30yr old self will thank you for doing it right.
You will read about people here applying for cards on schedules, and doing what you want to try and do. The difference is they have put in the time, have long solid histories, and a fee cards here and there doesn’t impact their overall credit picture to near the extent.
Also, don’t be tempted by crap cards with an annual fee (AF) that give you nothing and are from low-end lenders. When it is time, you want to start with good cards that have no AF. You want to use them at least every two months, if not every month. These are the kind of cards that you keep for a very, very long time. These are the ones that you build a very long history with, that will in time allow you to get a card here and there with much less impact.
It’s not what you want to hear, but it’s the honest truth. It sounds like you’re starting out alright, but don’t go and ruin it before you’ve really begun.