Burris low rings will have 1" scope tube about .130" above the flat of the scope bases. Going to weaver low rings will get you down to .090" above the scope base flats. If, howeveer, you are down at your jaw, chances are real good that the stock is dimensioned for iron sights. Use some sort of straight edge - lay your rifle on top with the straight edge in line with your bore. The measure the nose of the comb and the top of the rear of the comb to that bore line. Compare your numbers to a commercial rifle stocked for a scope - for example, a Ruger M77 Frontier is about 5/8" drop at nose and 3/4" drop at butt. A monte Carlo style will have highest part of comb higher than the bore line. Several mil surplus stocks are as much as 2" or more below bore line at the top of their buttstock.
Another suggestion - google up a picture of a Lee Enfield No.4 T sniper rifle and note that the attached a piece on top of the comb to get the better check weld on the scope which the rifle was not originally designed for. I have seen similar commercial butt "sock" type sleeves that can accomplish the same thing.