These are my 5 favorite roms that i have yet to use. Some give you fast experience and some also give lots of customization

 

5. CSRom 

https://phonetechkid.com/2013/01/02/csrom-for-att-galaxy-s3-2/

this rom is very fast and also gives a nice look to your phone check out my video review on it.

 

4. Urban Karnage Rom

https://phonetechkid.com/2012/12/22/urban-karnage-rom-for-att-galaxy-s3/

This gives you touchwiz and some 4.2 apps which makes this rom great. You also get KT Kernel and lots of other tweaks.

 

3. JellyBam Rom

https://phonetechkid.com/2013/01/02/jellybam-rom-for-att-galaxy-s3/

This mixes Paranoid, Aokp, Cyanogenmod, and Touchwiz. The Customization is endless.

 

2. Serenity Rom

This was the first touchwiz Jellybean rom and it makes your S3 have features the Note 2 has. it is a great and fast rom.

 

1. Illusion Rom (CM 10.1)

https://phonetechkid.com/2013/01/02/csrom-for-att-galaxy-s3/

This gives you stock android, jellybean 4.2.1, and alot of tweaks making your phone be as fast as it can.