I'm a 5 ft 10 in male and here is how I handle my bag:
I wear an Nu-Way support belt nearly 24/7 due to a hernia that cannot be repaired. The good this does for me is it keeps me looking trim and fit at the waistline! :) I'm only 177 pounds so it's not a huge issue to begin with but at age 55 there is a bit of muffin top. It's life.
Next, I wear an Ostomy Secrets wrap. This has a pouch for the bag to fit into and holds it secure against my body. I've come to find this combo reasonably comfortable, but way better than not using one or the other.
I also wear an Ostomy Armor support/protection belt at times (with the Ostomy Secrets wrap). It works great for protecting my stoma when I participate in my main hobby, racing cars.
I tried the clip-on plastic belt and found it just sucked, frankly.
I too have found, as others have said, the only person that notices your bulging pouch is you. I wear t-shirts, shorts, jeans, dress slacks, button downs tucked in, you name it, no problems.
Here is a tip: You can turn your bag 90 degrees so it "hangs" horizontally, then use a wrap designed for that kind of wearing the bag. I am sorry I can't recall the name of the company that makes those wraps for that bag-wearing method. It seems to work for active people as well. I do not use this method as doing so places the bag partly in the way of the seat and seat belts in my race car!
Hope this is of help, good luck and get out there!
Bryan