It doesn't necessarily need a deeper bottom if the mounting holes are higher on the cage (making the cage sit relatively lower).

This would be an issue of going to the bike shop and measuring distances, finding a cage that has "sliding" mounts (larger, elliptical, holes so you can position where you want rather than one fixed circular hole), or I believe they also may make an "adaptor" of sorts that would effectively change the position of your mounting holes (basically a piece of metal/plastic that has mounts for your cage AND another set of mounts that will mount to your bike and lower the cage).

Lots of potential solutions.

However, if you want to actually get that bottle in and out of the cage while you are riding you'll still likely want a side entry in addition to the fix because the fit will still be quite tight.

I use Arudel "sideloaders" http://www.arundelbike.com/products.html With no issues at all. Never lost a bottle. Unfortunately, because they are carbon, they aren't the cheapest out there. Hopefully they will soon make a "plastic" version of this because they are great cages. (Also, you'll note if you look at the close up picture of these that they are drilled for two mounting positions to help alleviate your current problem as well).

I manage to get two full sized (24 oz) standard bike bottles (a camelbak bottle would not fit, they are taller) on an Extra Small, sloping top tube, thick tubed, carbon frame. And I can access both of them on a ride. There are many a trick and it can be done