d20 assumes a simple progression of +1 bonus to hit for each HD. I was wondering how well that lined up with the original D&D TO HIT table. It's not quite that simple.
Looking at the Alternate Combat system (the top chart in gray) in the Original Men & Magic book the table is relatively smooth except in a few spots where it jumps. Between 2-3 and 3-4 hit dice the number needed to hit reduces by two. This also happens from 6-8 to 9-10 hit dice. We can take that as is and directly convert it to an ascending Armor Class combat system with a bonus to hit on d20. This is indicated in blue.
However with some tweaks we can smooth out those jumps a bit by spreading out 5 & 6 and then grouping 7-8 and moving 9 hit dice down to its own column.
I think these adjustments remain very true to the original To Hit chart while smoothly adapting it to an ascending Armor Class system.
The blue chart is useful for converting OD&D to d20 as well. For example a 4HD monster might be better as a 3HD monster in d20/5e.
Here are a couple other To Hit tables from early DnD that ever so slightly smooths out this progression by delineating the overlap a little more clearly, like separating 3 from 3+. From the Holmes "Blue Book" basic:
And this classic from the indispensable, hugely useful, Judges Guild Ready Ref Sheets which appears to be identical to that in OD&D, just expanded.