This is a great, great question. It doesn't seem like a big deal, but it can be a real challenge to solve. What's more, if you're backpacking, you have no more room for much else, even with a 50-75 liter pack. What's a soundie to do?
I pick a pack that will carry my essentials for the day/afternoon/hour and will also fit at least my mics in their zeppelin. The problem is that that $#!% gets big. When I'm doing serious backpacking, which is the worst-case-scenario for carrying a lot of stuff in addition to audio gear, I shove the zepp horizontally underneath the lid of the pack and I wear my SD 702 in a Portabrace with a chest harness lifted from a LowePro TopLoader SLR bag. Fits great and backpack straps go right over it. (Or, I just bring a handheld recorder if the conditions and subject matter warrant. Way easier/lighter!)
Never overlook the ability to pack things INSIDE the zeppelin/blimp/windscreen - great way to carry a fleece or other jacket, and keep the mics nice and snug. Lots of room in those things. Less to worry about if rain hits the outside, too. I've successfully loaded up a backpack and a messenger bag (both Chrome brand) with full kit, but with not a lot of room for the essentials. Gets crowded in there with a zepp holding an M/S rig!
Sometimes I attach the windscreen to a boom pole just so I have something to strap to my recorder case, which is easier with a Petrol case than a Portabrace.
Your idea of using a coat to cover the zepp itself lets you strap it to the outside of your pack. Grosgrain-webbing compression straps can be purchased cheaply, or you can get a few meters of shock cord with a cord lock, depending on the attachment options of your pack itself. You can also look for silicone-impregnated nylon (silnylon) stuff sacks at outdoors stores - not waterproof but reasonably splash-resistant, better if you seal the seams with McNett SeamSeal or a mixture of silicone caulking and mineral spirits. My local REI sees me arrive with audio gear all the time. You'd be surprised how similar windscreens are to rolled-up sleeping bags in size and girth.
I've seen people with smaller rigs (like a PCM-D50, Sound Devices MixPre, and lavs) operate out of fanny packs or even large jackets with lots of pockets. Gordon Hempton rolls out of a Domke photog bag (I have one, they're incredible, but a zepp often won't fit inside). Others jam stuff in backpacks, which is far more common.
Air travel's totally different, and actual mic cases, pvc tubing, and Pelican cases are great for serious transit - but I'm assuming from your question that you're talking about mic carry on-person.
I've had some recent huge successes with some unusual bags that I'm still experimenting with. Formal reviews shall be forthcoming in September! Stay tuned...