From my testing over the last 10 years with amateurs in my schools and professionals on the PGA Tour the saw grip consistently provides superior results for putts inside 10 feet and good results for putts from 12 to 20 feet. But performance on long putts tends to be worse than with conventional grips.
Here’s your homework: On the green make the last couple of practice strokes with this (and only this) in mind. Focus on matching your speed to the break you read. You’ll start pouring ’em in from all over. And remember lip-ins count the same as ones hit dead in the center of the cup.
It’s the same series of steps I followed for the putt in the photo at above right—a 25-foot downhiller on the seventh green at Shinnecock Hills. I read four feet of break from right to left. As such I aimed four feet right of the hole. After making a few practice strokes to dial in the correct speed I putted.