I've ranted before about the need to have some sort of blow-out capability on your person all the time. Thankfully many companies are producing an ever-evolving line of good, low vis, simple to carry TQ carriers or blow-out kit which I believe everyone should carry daily (whether you carry a gun every day or not). Many have signatures less obvious and intrusive than an OWB magazine pouch. One option good for anyone is the Eleven 10 - Kydex C-A-T® Tourniquet Case Slick Front. It's actually only one of several options manufactured by Eleven 10 (a veteran owned company with products made in the USA). Another is the Eleven 10 - Kydex SOF®TT Tourniquet Case. There are several others and all retail for less than $40.
The slick front C-A-T TQ Case holds C-A-T® Tourniquet, can be mounted to a duty or daily wear belt vertically or horizontally and is MOLLE-capable. The case can also be mounted to a Safariland Drop Leg Shroud using the exisiting holes (the case mounts with 2 screws in the middle of the case). Kydex holds the TQ in place without retention strap or cover but does not impede a fast retrieval, which is accomplished by pulling the exposed part of the TQ.
Want something with more than a basic TQ? Get an Eleven-10 Personal Med Kit/Blow Out Kit (which comes filled with all the first-aid rendering goodness you might need for an exigent response). It comes with a Tac Med Solutions SOFT-T Wide Tourniquet, Tac Med Solutions Olaes Modular Bandage 4",QuickClot Combat Gauze LE, MedStorm Nitrile Black Exam Gloves - Size L and Eleven 10 Small Personal Med Pouch.
Look too bulky for you, want something to just slide in a back pocket? Consider the Pocket DARK from Pocket Doc of Dark Angel Medical. I actually have two because I invariably lose track of one. You can get it in a black or coyote carrier, or grab on in purple for the soccer mom van. The Pocket DARK measures 4" x 4" x 1" and weighs in at 7 oz. It is packaged in a vacuum-sealed 4 mil thick bag that could also do double duty as a field expedient occlusive dressing and comes with a SWAT-T and z-folded QuikClot Combat Gauze LE. MSRP is $80.
Prefer something put together a little differently, with more options for belt carry IWB/OWB and different kit inside? The Rogue Gunfighter IWB low vis med kit is supposed to shipping next month and Raven Concealment Systems has a new low-vis blow-out kit they made in collaboration with Eleven-10. It should be out in a month or two, perhaps you could ask the Residential Sergeant Major for an RCS gift certificate.
This is only a limited look at the options out there. There are many others and given the wide variety it's a given somebody somewhere has one that suits your needs. Hell, Tuff Products has an ankle-carry TQ holder that's perfect for people whose job require them to wear long pants even now that it's summer.
Whatever you decide on, carry something as you go about your daily life (and damn sure carry one on the range). Make sure you're familiar with the contents and how their stowed too - practice with your med kit the same way you'd practice a reload or dry-fire. Maybe even with the same frequency, after all...which one are you more likely to use, handgun or med kit?
I'll leave you with this. Take a few minutes to listen to Jim Gilliland and Kerry Davis. They talk TQ application right after the introduction from Mike Lamb (who has gotten to be almost as big as I am).