I have an iPhone 6 and just did the iOS update a couple weeks ago, at which point I was having this problem too, and after about an hour of reading through these posts and trying everyone’s suggestions with no success, I was about to give up. I had one more idea and it seems to be working. Worth a shot.
In Sounds settings, my toggle for “change with buttons” is turned on. Then I used my volume buttons to set the volume to the level I wanted for my alarm. Like you said, once I hit snooze, the second alarm would blare. So, my last idea was to keep the toggle button on, reset the volume back to the level I wanted, and then in the screen below, I noticed the volume level was much higher, so I adjusted that to the same level as I put for my alarm, and tested it out. My alarm went off and the volume did not go up for the second alarm (after snoozing). The ringer and volume bars still pop up on the screen without me doing anything, but the volume level does not change. Hopefully this will help others having the same problem!!