Are you sure these are Fico scores?
It’s very unusual for someone with just one credit card and a short credit history of just one year to have such high scores. Where did you get them?
I’m not sure about TransUnion, but Equifax is no longer selling Fico scores to consumers. They only sell Vantage scores now. A 711 Vantage score is pretty mediocre, which is where I would expect your score to be with a very short credit history (See credit-score-range.html for a comparison between Fico & Vantage).
The ONLY place that sells actual Fico scores to consumers is myFICO.com.
Having 4 hard inquiries is not brilliant, but it’s also not as bad as you might think. Each hard inquiry will only lower your score by a few points. They stop counting in your Fico after 12 months, and completely fall off your credit report after 2 years. Of all the things to worry about on your credit report, inquiries should be at the bottom of the list.
If you’re not in a hurry to get a reward card, I’d wait another year before applying. 90% of your credit score is generated by the last two years credit activity, which you lack. You will have a much better credit score a year from now.
After a year, check with your bank and see what kind of reward program cards they offer.
Remember that score is not everything, so you may not qualify even with a higher score. They may look at your debt-to-income ratio or other parameters.
Try Discover card. I hear it is easy to get approved on.
Good luck