You didn't mention it. But I assume you've been to Piazza Navona, walking distance from your hotel, a few blocks from the Pantheon. If you've been there try Trastevere. A very old and quaint neighborhood with ancient streets and beautiful plazas. What we call the Tiber River is called Fiume Tevere in Italian. Trastevere means across the Tiber River. Not near the Vatican but down the river more towards the Colosseum. You can't leave Rome without seeing Piazza Navona, and if you've done that, consider Trastevere.

http://www.parlafood.com/katie-parlas-rome-trastevere/

PS: it seems that you are finding out that traveling to Italy and enjoying one main spot is very satisfying, and is much better than trying to go to as many famous cities as possible to take as many pictures in front of famous buildings as possible, and leaving not knowing very much about italy.