Supermarkets can spawn tents, pretty much one of the few places that tents can spawn IIRC.
http://dayzdb.com/map is a handy map to check on spawn locations, seeing as you are at Prigorodky, you could try to go into Cherno or Elektro and risk getting your head blown off, of you could hike to a smaller township and attempt a scavenge there. And when I say hike, I do mean a hike
From a wiki list of supermarkets
Berezino which is the only town to have two.
Chernogorsk
Elektrozavodsk
Krasnostav
Polana
Pustoshka
Stary Sobor
Vybor
Zelenogorsk
If you strike NW you can try Zelenegorsk, Pushto and Vybor, North has Stary Sobor (expect death), NE leads you to Polana then Berenzino for 2 and northwards to Kras. You could try for a tent in Cherno/Elektro, its just that on the server I play on, two of the crew that I hang with were sniping noobs in elektro for lulz.
The other night I went looking for my pushbike that I lost thanks to a server reset (server I play on doesnt announce timed server resets, just resets every 6 hours, really fucking annoying if you didnt get out and save your vehicle recently, means you ahve to leg it from where you were at time of crash to where it was last saved) anyway, looking for my bike in the woods for an hour or two, finally decide to give up as ive ran out of water when I decide to head to the nearest town to try and snag a soda, see something in the trees thats a little too solid for a shrub. Found a motherfucking hatchback 400m from the road. Jump in and check it, all green with fuel and a jerry in the boot, off I go. Taking it back to base camp I run along some shitty dirt road (car is slow as crap on dirt) as I get near the highway I see the Military ural in front of me, get on the road and hammer it to base thinking of picking up both of them (I know, massively retarded choice to keep on the shitty hatchback instead of upgrade right away, decision making at 1am isnt the best) anyway as im banging down the road about 1k from where the ural was server decides to reset...
kinda pissed at that I quickly realise that im really close to that motherfucking ural, nobody but me and the owner know where that hatchback is stashed, and I can now get both. So as soon as its up I hump back down to the ural, hop in and its passable, engine is a bit red but everything else works so off I go cross country. Manage to get it safely back to base and check its gear. Shit tier AKM's makarovs and shotguns, take out its wheels with my silenced M9 as it trundles down the hill originally parked next to our civilian ural it ends up a distance away now. Then I go fetch the hatchback. Last count of vehicles is 4 ATV's, 2 Off Road Trucks, 1 Gaz for speedy highway travel, 1 Hatchback, 1 Civilian ural and 1 Military ural for storage. Never found my bike but my bro went and got his pushy from where he crashed it last. So many vehicles is a liability but its also so very damned handy for getting around and storing shit.