The problem with you budget is you are deep in Junk car territory. You may be lucky and find something that will last several years but you are also more likely to find a car that needs lot of money spent to keep it running.
Also at this point the previous owner is more important than brand. Ideally look for a car that has been owned by an elderly person. Also look for cars that are generally reliable but aren't popular - for instance early 2000's cars from an orphaned brand like Pontiac, Oldsmobile or Mercury. Look athttp://longtermqualityindex.com for idea. Steve Lang (who is behind the about site) has written on many site, for instance
Make sure whatever you buy you get an inspection from YOUR mechanic so you know what issues there might be.
However before you go down this rabbit hole, what about this car that you recently brought? How much would it cost to repair? Was it brought from a dealer? When you brought it did it have any powertrain warranty? Was the failure an issue you caused or an underlying problem that was there when you brought it. What State are you in?
At the end of the day if you buy a $1000 car and have to spend $1000 on tire and repairs to make it road worthy you'll still have a car worth $1000, however if you put that $1000 into repairing you current ride you have a car worth Much more. Make sure you get a quote from several independent mechanics.