Reddit mentions of IKEA Flyt Magazine file, Pack of 10, White

The easiest and cheapest way to sort and temporarily store paper would be to have it vertical, especially if each of the current stacks is its own category. You'd be able to use much more of the existing shelving unit and that would mean whatever isn't making it into the closet right now might fit. I think some magazine holders like these from Ikea would be a big help (Also sold at Amazon.)

Sort the cases by whatever makes the most sense: Last names, dates, etc. I'd use post-it notes on the magazine files until you are certain what kind of labeling structure works best. You won't want to decide to change it halfway through permanently labeling everything.

I hope this gets you started!

In addition to the Thrill Containment Units already mentioned, these also work decent enough if you tall a shelf: https://www.amazon.com/gp/product/B00SK0V466/ref=ppx_yo_dt_b_asin_title_o07_s00?ie=UTF8&psc=1. I use them for both 2000ad and megazines.

...magazine caddies range can dramatically in durability and price, but after a lot of shopping around i eventually settled on ikea's flyt caddies: they're a little bit thin, but they work fine for this use case and were reasonably cost-effective when ordered in bulk, about two dollars apiece...

https://www.amazon.com/IKEA-Flyt-Magazine-file-White/dp/B00SK0V466

...it looks looks like ikea have since discontinued all their cheap cardboard caddies; flyt is no longer listed on their own website, but the similar fluns model is on clearance for crazy-cheap, 25¢ apiece!..

http://www.ikea.com/us/en/catalog/products/00324132/