What type of base do you have on there now?
Burris Sig ZEE rings on a weaver or similar base will resolve your issues AND keep that scope low.
Many MOA bases are by nature quite thick/tall. This then requires lower set of rings or a scope that might be too high for a proper cheek weld on the stock.
A very nice set of steel bases are made by Burris in their XTB line up. Well priced and similar sizing to the generic Weaver alloy base (Weaver bases will also work just fine).
Add on a set of Burris Sig ZEE Meds and you are off to the races. If you have a large bell scope, then the HIGHs will get the job done.
Only having 12 mins of up is in no way an indication that something is wrong. When a scope ring set up is 'perfect', the elevation should be 1/2 the internal amount or in the CENTER of scope travel. So you are 8mins off center - Big deal.
Shimming a scope puts your 100yd zero at the bottom of the scope travel.
If you go with the Burris rings, I would add 10 thou and see what happens.
Because of the differences in height and what the scope will now 'see', shimming is not always a linear adjustment.
But with these inserts, you can play around to get whatever 100yd zero that makes sense for what you are doing.
In general, you do not want a scope to be sitting on the bottom or top of its travel - degrading optics and zero shift may occur with some scopes.
You should be able to get to 1000yds with 25 to 30 mins depending on the bullet you want to use so you don't need to go to the extremes of travel.
If you dont mind using the mildots, you can reach that right now.
Let me know if I can help you the products to set up your rifle.
Enjoy...
Jerry