I know there are more experienced public wearers than I on this site, hopefully, they will chime in as well as I think they can be more practical with their advice, but I have worn outside of my apartment on more than a few occasions and I have never been caught.
I actually wear a pair of underpants and baggy clothing over whatever diaper I wear. Underwear I find reigns in the puff and makes the diaper bulge a lot less noticeable. If you slide a pair of baggy pants over that, no one will be able to tell anything. This two part system, underwear over the diaper and baggy pants has worked wonders for me thus far. I only wear outside of my house if I am heading to an AB or a DL friend's place. I don't worry about the bulk of my diaper in these situations because like I said the underwear and baggy pants combo really reduces the puff and it also curtails crinkling as well.
I also think a large part of being comfortable wearing out and about is mindset, you just can't worry or be paranoid about someone finding out. This is easier said than done, I was nervous when I wore out to friend's places the first few times, but I realized that I had the walk down, my underwear and baggy pants system really did conceal it well and most people you bump into on the street are either A) not going to notice or B) notice, but not be bold enough to say anything out loud.
In answer to your thinner brand question, there are some thinner diapers out there that some of my more 24/7, wear to work pals use. Some of them use just regular store brand briefs as they are awfully thin, but they aren't very functional, although, it doesn't sound like you'd be using in public, just wearing discretely underneath your normal clothes. I've also heard that ABU Pre-Schools are thin in comparison to their other brands making them a good on the go diaper. I have also heard good things about Pull-Up/Pull-On style briefs like the ones offered by North Shore or Bambino.
Hopefully this helps and like I said, I'm sure you'll be having a bunch of more experienced out and about wearers offering their advice as well