How can I fix my iPhone that won't get out of restart screen?

I have an iPhone 6 that I've had since October or November 2014, with 64 gigabytes of storage, iOS 9.3.2 I've never had any major issues with it. However, two days ago the phone restarted randomly. I didn't think much about it until yesterday, when the phone froze for a few moments, went black, then went to the restart screen (white screen with black Apple logo), went black again, went to the restart screen, black again, restart screen, and then the lock screen. I could get into the phone at this point. It did this process several times, finally showing the plug into iTunes image, which I did. Although I already have iOS 9.3.2, iTunes said something about needing to reinstall it (or at least that's how I understood it), which I let it. I also made a backup of the phone. After that, it worked fine.

Today, the phone froze, went black, and showed the restart screen. It doesn't get past this screen, although after several minutes it goes black and doesn't wake up again until I hit the power button, in which case it does the same thing. If I try a hard reset on the restart screen, the phone goes black and then instantly returns to the restart screen. If I try a hard reset on the black screen, it goes to the restart screen. iTunes doesn't recognize the phone, so I can't reset it.

I haven't done anything big with the software recently, except turn on and off the phone over wifi feature and delete a few unnecessary apps. Apple Support couldn't help and recommended that I go to an Apple retail store, but I am unable to for at least a week.

Other (yesterday): The phone either crashed on the home screen after I logged on or on the app Bleacher Report's Team Stream, although I don't know if this is the app's fault or if, as it is the app I use most often, it is just more likely to crash there than on any other app. Today, the phone froze on the Team Stream app and I haven't been able to access it since.

Does anybody know how I can fix this problem? I have some data that I have not backed up to the Cloud on the phone, and I'd rather not lose it by resetting the phone and not reinstalling a backup. Thanks in advance.

Sorry if this is the wrong section to post in, but I can't seem to find any other that fits.