A shredded tail is tail rot so you diagnosed right. It is often caused by dirty water or sitting on gravel that has lots of "yuckies" on or in it. How big is your tank and how often do you change your water? Also, how much water do you change at a time? Make sure that you are doing weekly water changes and that will help with getting your fella's tail healed up, too! If you do NOT have a filter, change 75%-85% of the water weekly. If you do have a filter, change 25%-50% of the water weekly.
**NEVER DO A 100% WATER CHANGE. This will only dump any and all beneficial bacteria down the drain which helps to keep ammonia and nitrIte spikes from happening. Okay, now that we covered that, what to do about the actual tail rot itself.
First of all, the Melafix is a good med to use especially if it is a new case of tail rot and one that is not too serious. However, Melafix doesn't do well killing the bacteria causing the tail rot so if you get a more serious case of it you'll have to more to something stronger. Along with the Melafix, add some AQUARIUM SALT to your tank. Add 1/2 teapsoon of salt per gallon of water.
**Dissolve the salt in conditioned water BEFORE adding it to the tank. The salt will help your Betta produce a thicker slime coat thus adding to his protection and helping him rebuild his tail.
Remember that salt doesn't evaporate and isn't filtered out so be careful to not oversalt your fish. If I'm adding salt to my tank, I only add it every 3rd or 4th water change. All of this should help, but if it doesn't work, the next thing you can use is Maracyn Two. Follow the directions on the package exactly.
Remove carbon from your filter with any med you use or you'll only wasting money and meds. 
The Maracyn Two will make the water a bit cloudy but that will go away and doesn't hurt anything. Also, make sure that you have water agitation with the Maracyn since it lowers O2 in the tank's water.
Hope this helps! Let me know how your Betta is doing and if I can answer any other questions!! Good luck!