I learned this lesson the hard way. I had the same problem that you are having only this was an important shot and the person wanted an 8x10 not a 8x12. When I took the picture I framed it too close on the sides and when I went to make an 8x10 part of the people were cut off. When your camera is set for 4x6 prints and you make an 8x10 print you will lose one inch on each side.
The aspect ratio for 4x6 prints is 3:2. 4 X 2 = 8. 4 X 3 = 12 So you get 8 X 12. (When specifying prints you always give the short side first.)
The aspect ratio for 8 X 10 prints is 4:3. It will not come out mathematically like above but that is the aspect ratio for 8 X 10.
I don't know if you can change the aspect ratio in the menu of your camera but I know I can on my 5D3.
So if I know that I want to have 8 X 10 prints made I can set my aspect ratio to 4:3 in the camera menu and frame as tight as I care to.
Or I can leave it at 3:2 and ALWAYS remember to shoot WIDE knowing that I will lose one inch on each side when the prints ( 8 x 10 ) are made. When I shoot wide I import them into Lightroom and crop them with the 4:3 (8 x 10) aspect ratio and I know exactly the frame I will get when they are printed as 8 x 10.
Live and learn. I usually do more living than I do learning!
Note that 10×8 is NOT a 4:3 ratio but a 5:4 ratio, so you would still need to crop a 4:3 ratio compact/micro4/3 image, just not as much. I think the most commonly seen 4:3 ratio size is actually16×12.