Each time you increase the base number by 2, you increase the average by 1. So going from 1d6 to 1d8 is an increase of +1 damage (in this case, going from 3.5 to 4.5 average damage.)
Going from 1d(2n) to 2dn is slightly different. E.g. 2d4 is slightly more damage on average than 1d8, but the hardest hits are much rarer than the average hits as a result (8 has a 12.5% chance on 1d8, but only half that on 2d4.) It's still realistically an upgrade, but it's a smaller one.
If you want to go into higher die sizes, you'd have to accept partial measures. E.g.
1d12
2d6 (a weak boost)
2d8 (a big boost)
2d10
Or, alternatively, you could go from 1d12 to 1d6+1d8, which is a bigger boost than average. Then it goes up one die at a time, 2d8, 1d8+1d10, 2d10, 1d10+1d12, 2d12. At that point, you shouldn't have to worry about any other results anyway, because you're already 6 steps beyond the maximum damage die size in D&D proper.