This takes a bit of explaining...
There are essentially 2 ways to get from Leece to Ugento:
1) Salento in bus: but this service last year started end June. so you have to look a few weeks before you leave to see if they reinstate it earlier. http://www.salentointrenoebus.it/presentazione.html
2) Train/Bus. http://www.fseonline.it/ Here you can download the time schedule. There is a small exchange due to an interruption of service... so sometimes you start with a bus and then exchange to a train at Carmiano -Magliano. The time schedule is last year's, but it will more or less remain the same (you will have to check closer to your departure date)... so in your case... there's a service at 16:45, 18:48, 20:40. And return 8:07, 9:11.
I will send the schedules that I was able to download, and enjoy your holidays!
Now, I realize that we are all trying to help people on this forum. But it would be nice if people would refrain from making general comments that are not true. There is Europcar which is a main Car rental service company that has an office in Ugento. http://www.europcar.it
Heather, I do not know your exact schedule... a piece of advice... it would probably be more convenient (in terms of times and flexibility) to rent a car at your initial arrival destination in Puglia (maybe Brindisi Airport or Lecce Train station) and drive to Ugento, as the train service (as you will see from the schedule does a huge amount of intermittent stops... By train/bus Lecce/Ugento is about 1.5hrs and by car, Lecce/Ugento is about 45mins.
Let us know if you have any further questions.