Just saw a prepper who added a full trauma kit carried around his waist; concealed and lumpy.
At some point, just get a bag - a laptop case, fanny pack or backpack. You will have more room.
One of my friends keeps a full trauma kit in an ankle rig. He wears it every single day, whatever he's doing. To my surprise he's been doing that for well over a year and other than minor adjustments it seems to work for him.
I keep a TSA-safe trauma kit in my laptop bag among several other places.