Unless you have someone add you as an AU on a 6+ months tradeline, You should be able to see FICO score(s) in six months I think you should consider opening a second secured CC. (Maybe Wells Fargo or Bank of America?) Lenders prefer to see more than one tradeline in your report(s) and i believe if you get two CC right now, in six months you'll have a good shot getting the Discover it
Many banks like Chase and Bank of America don't see store cards like real cards cuz of the fact you can't use them anywhere so i believe a second secured card will help you more than a store card
Greetings Charmlessman!
I have had a long credit journey and a long history here on the forums as well. If I were you, I would just sit back and wait for now. Restrain yourself from apping for anything.
After about 6 months or so, the inquiries you currently have will not hurt you as bad and your history,albiet limited, will mean more.
Be careful about what you app for. In my opinion, store cards have a purpose, but even the master card or visa store cards can be tricky. Those can be used pretty much anywhere, which makes them easier to just pull out & use for anything. They also have interest rates of about 29%, so if you carry a balance, thst will cost you more.
You might be better off hanging onto that secured card for a while...add more to it. Get a bigger limit going on it & after a year, ask for a product swap...go from secured to non-secured. This can likely be done with no hard inquiry. After the swap, that is when I would app for another card.
In the meantime...I highly suggest joining a Credit Union! They have lower rates, better products & they are owned by the members...not a huge banking institution with fat cats at the top with fat salaries! Credit unions are also more forgiving of those with a limited, or not so perfect history.
I know you want to get going on your credit, but be careful. Do some research & in the end...app for the product that you're likely to be approved for, not the product you simply "want". They are often, not the same thing.
Oh.....one more thing.,...Welcome to the Forums & Welcome to America!! 
Starting your US credit journey really depends on how much income/cash you have. Unfortunately, as with everything in life, money allows more freedom in getting things done that help.
The thing that would have helped me the most was to have an AMEX card before arriving in the US. You could then use the global transfer program and get a US AMEX issued instantly. Unfortunately, AMEX cards are pretty much useless anywhere else so I closed mine a long time ago. That would have saved me time and money. Oh well.
If you can't do the AMEX thing, opening a secured CC at a credit union is the next thing you should do. Check out SDFCU or your local CU. My local CU had a great secured product - Visa platinum, no FTF, no fees, really low APR and a decent 1% cash back program. Put as much money as you can afford to lock away into securing the card (the optimal amount is $5,000. Locking that away is a bit rough on anybody, so try something like $3,000 if you can). Some CUs will open a secured CC without an INQ.
Once you have your card, PIF every month for 6 months. Right before the sixth month gets reported, do whatever is required to get it to report a util of 1%. You'll now get a score on Credit Sesame. When a score shows up in CS, you now have a FICO generated (though your CS score isn't a FICO score, blah blah blah). At this point, a clean file with 6 months of history and 1 TL with a CL of 3000-5000 and 1% util will net you a EX FICO of 750. Congratulations, you now qualify for any card on the market, except for one small problem. You only have 6 months of credit history. 
There's been some talk of Chase issuing Freedom cards at 8 months, but they will definitely not issue a CSP or Ink at 6 months. You need to wait until at least 12 months of history have reported. Your only real prime option at this stage is AMEX.
With a high enough income and an EX FICO of 750, AMEX will issue you a charge card (the PRG is particularly good for the first year). Once that happens (and you can actually do this a minute after you get approved for the charge card), you can apply for a revolver like the SPG (also no AF for the first year). AMEX will collapse both INQs into one for both cards. Your CL won't be great (roughly the same as the highest CL you have) and your APR won't be great, but it's a start as your first unsecured CC.
Wait 61 days, do the 3x CLI. You now have over $10K on a TL. That's a good thing because if you get a CSP later, they'll probably match that. You should probably garden for at least 6-12 months from this point.
Whatever happens after that is up to you and your income level. Of course, YMMV with all of the above.