It's an Alcatel "One Touch Pop Icon" using OS 4.4.2 Kit Kat.
The apps I've tried don't seem to get it to a lower setting. I never use head phones. I doubt it's a problem with just my phone. I think it's working as designed. It's just too loud in a quiet setting.
Is there a way to safely dampen the volume at the speaker? It's in a case so I suppose I could put something over the speaker but I'm concerned that simply putting some tape over it might mess with the speaker's engineering.