How do you organize your academic papers?
I highly recommend a software to manage this like Endnote, Zotero, or Mendeley. The last two are free if your library is small enough (depends on the size of the PDFs). That will make things much easier by cleaning up the metadata and helping you use search tools to find the right papers.

In each of these software, you can also sort the papers around. The key is that within these software, you can put a paper into multiple folders/categories/groups/whatever else they call it. I use this liberally. Note that the software will keep the reference as one reference with multiple links, and so if you do things like annotate or add notes to the file they will be visible at all locations where you can find the paper.


Before doing anything make sure the metadata is all correct. You can always last resort with using the search bar if it is.
First put each paper into a field. So I would make a folder like breast cancer, and then have a subcategory like metastasis if I have a bunch of papers specifically on that subtopic. Finding odd papers reduces to remembering what subtopic it's from, and finding that subtopic folder within the right topic.
If a paper works in multiple different places, I put it in each place.  
Then add the papers to project folders. I make a master folder for each project, and subfolders for different parts of the project, put the appropriate references. The software can now automatically generate a file with all of the references for the paper which helps with citing.


Using this method, you can always dive down into a folder/subfolder that makes sense, either by project or by field, then sort by title/author/year/etc (and easily switch between these), or use the search in the subfolder, and always quickly find what you need.

But... you will need to take a day or two to set this up. That's a long and tedious task, but definitely worth it. Then I usually just do a minor fix up for new additions every few months to keep it tidy.