Recover comp that I accidentally saved over

Put "comp" in the search field of the project panel, this way after effects will list all the compositions in the project, take a look and see if you can find your composition, maybe you put it inside a folder by mistake:

If you have autosave turned on, and everybody should, there will be a version of your project in the After Effects AutoSave directory. 

If you do not have autosave turned on and you don't save versions of your project, again, everybody should, then you are out of luck.

I'm confused by your information.  You say you started the newproject in another comp.  if that's true then both "comps" would existed in the same project.  So everything would still be there.

On the other hand, if you mispoke when you said you started it in a new comp and what you really meant ws that you started a new Project.  Then that would mean that you'd never saved your first project with a unique name.  When you start a new project in AE it by default has the name "Untitled Project".  If you quit or start a new project AE will ask you if you want to save the current project.  The only way I can see your post making any sense is if you are saving all your projects with name "untiltled project" in the same folder and just overwriting every project all the time.  In which case, you need to go back to baby steps and learn how to use AE.  I mean no offense but The way you wrote it is like "another small project popped up" makes sound like you are working in some official capacity with AE.  If that's the case I don't understand why you would start a new project inside of an existing project.  And if you did, why would have had "opened your AE file again to resume your previous work"?   None of it adds up.  Are you sure you're correctly explaining what you did?

We can't know what you don't know. It's as simple as that, even mnore so since yoou have not offered evene the most rudimentary system info. If you use a Mac, there may be a realtively straightforward way to restore an older file version, but otherwise there pretty much nothing your can do. Restoring an overwritten file would require to use some pretty nerdy stuff and even then it's not guaranteed that the resulting file is even usable.

Mylenium