There are a handful of... I guess you could call them upside-down towers, where you can use Ascend to end up on the surface. They're illuminated all down the column, so you can see them from afar, and it's quite obvious what they're for when you see them, but there are only a few of them. I'm pretty sure there are a few parts of the Depths that don't have one and are walled off from the rest of the Depths; those you won't be able to exit without some other method.
As an alternative method, you can use the powered floating platform zonai device in conjunction with balloons, fans, or rockets to go up, taking out another floating platform when the one you're on is about to expire.
(spoilers for a mechanic you may not have unlocked yet below)
By using Autobuild, you can just make floating platforms wherever you want. I checked, and Autobuild will not use something you're currently standing on to make a new construction, so you're safe from falling off when you make a second platform. (This answer previously recommended alternating between small and large floating platforms in case it did pull the one out from under your feet.)
You probably want to use the small powered floating platforms, not the large unpowered ones; the large floating platforms are by far the most expensive thing to make with Autobuild, costing about 100 pieces of zonaite to create. They do give you plenty of room to work without falling off, though.
It may also be possible to climb up the walls of the chasm, if you can reach it with a simple balloon-and-flame-emitter combo. Since chasm walls are covered in gloom, you'll need lots of sundelion and stamina-restoring food, and gloom-resistant clothing wouldn't hurt either.
Even if you can't fit enough food in your inventory (there is a limit), you can of course use Autobuild to generate a safe platform to stand on, using a zonai stake and some kind of wood, metal, or stone slab, and then use portable pots to make more. Just stock up on sundelions and endura shrooms first. (Remember, a stamina overfill effect gives you a full stamina restore as well! So even a single endura shroom or endura carrot can give you a full 3 wheels of stamina, plus a little extra. Much more efficient than stamina elixirs.) Even just the platform alone would allow you some time for your gloom resist to recharge, too.