Joshua raised some good points in his comment to my earlier post (Horns of a Dilemma). At first I was going to comment back, but the comment was getting involved, so I decided to make it a full-on post.
A point I failed to make clear is that the additional attacks I proposed at 4th and 8th levels in no way combine with the F-M's ability to make one attack per opponent if all opponents are 1HD or less. That ability, at first glance, seems to scale quite well as the F-M's levels increase. On closer inspection, and in a more realistic light, I don't think it really does.
My proposal is to increase F-M's number of attacks at 4th level. Well, by 4th level most Fighting-Men have "outgrown" 1HD monsters. Their adventures and foes have gone beyond 1HD creatures, by and large. The treasure they carry is paltry compared to what 4th level adventurers can expect, and the XP for defeating them is only 1/4 listed value. So, not only are the 1HD creatures simply not encountered very often, they are undesirable as opponents when they are encountered. That is an "ability" that will see less and less use as the F-M rises in levels. That is a far cry from an ability that scales with level.
As the F-M goes up in level, I believe his ability to slay stronger foes should rise as well. By 4th level he will begin encountering creatures that deal more than just d6 damage, either with a bonus to damage, or with 2d6 damage. There aren't a lot of such creatures, but they are out there, and F-M need to be able to stand up to them. This ability needs to be represented by more than just an improvement in the attack tables. Since creature AC generally improves as the creatures get tougher, that's a wash anyway.
Another thing that I feel somewhat balances this issue is that the F-M is having to make two separate to-hit rolls, whereas the monsters that deal 2d6 do so with a single roll.
There was one other ability I would give Fighting-Men, that I forgot to mention in the other post. At 4th level they are able to hit opponents that require a +1 weapon to hit, and at 8th level they are able to hit opponents requiring +2 weapons. That comes from Chainmail, as well.