Lost files on hard drive after Time Machine back up
First things first, switch off Time Machine. Any further backups to that drive will hamper your ability to recover your files. 

You're going to need some sort of data recovery app to dig down through your hard drive to recover the data that was there which has hopefully not been overwritten. 

EaseUS is a pretty good recovery program that I have used to recover data in the past. Their Mac version is a little pricey, at $89.95, but you will have to gauge the value of your data you need to recover and see if it works out worth it. 

I doubt you will find any decent free application that can properly recover your data. EaseUS do have a trial version available that will allow you to recover up to 2GB of lost data so you can try it out risk free.