When I recently played in Evan’s Uz campaign, he had hit dice do double duty as attack bonus. Uz is based on Swords & Wizardry WhiteBox, which, like OD&D, only uses the d6 for hit dice (modulating the difference between classes using bonuses as described here). The elegance of this approach impressed me. Only one number is required for both hit points and offensive ability. And it generalizes to monsters, though you might want to cap the bonus depending on the campaign power curve (max +10 seems pretty reasonable to me).
I don’t remember exactly how he did it, but this is how I might do something similar. Fighters find their attack bonus by adding their hit dice expression together. For example, a fighter with HD 5+1 has a +6 attack bonus. All other classes ignore the bonus part of the hit dice expression and just use the base HD. So, a cleric with 4+1 hit dice attacks with +4. Tougher classes always have more hit dice than weaker classes, which is also how attack bonuses should work.
This is what the attack bonuses would look like given the recently posted rationalized hit dice progression (hit dice are in parentheses):
|1||+2 (1d6+1)||+1 (1d6)||+1 (1d6)|
|2||+3 (2d6+1)||+2 (2d6)||+1 (1d6+1)|
|3||+4 (3d6+1)||+2 (2d6+1)||+2 (2d6)|
|4||+5 (4d6+1)||+3 (3d6)||+2 (2d6+1)|
|5||+6 (5d6+1)||+4 (4d6)||+3 (3d6)|
|6||+7 (6d6+1)||+4 (4d6+1)||+3 (3d6+1)|
|7||+8 (7d6+1)||+5 (5d6)||+4 (4d6)|
|8||+9 (8d6+1)||+6 (6d6)||+4 (4d6+1)|
|9||+10 (9d6+1)||+6 (6d6+1)||+5 (5d6)|
|10||+11 (10d6+1)||+7 (7d6)||+5 (5d6+1)|
Another way to look at this rule would be that all classes use base hit dice as attack bonus, but fighters get an additional +1. Or, one could just use the additive hit dice for all classes, which makes things simpler (no special case for the fighter) at the cost of decreasing the relative power of the fighter slightly.
Using this system necessitates running with ascending AC. I have actually been considering switching to ascending AC off and on for a while now anyways. This attack bonus system is probably simpler and easier to understand than my attack ranks system, and it does away with another table (usually a good thing).