I don't know if a stronger Overlord was possible in the short term.
IIRC landing craft used for Husky were sent to the Far East for the Burma campaign. If they are retained in Europe then landing more troops in the first wave becomes possible. The early 1944 revisions to the plans only increased the first wave from 6 to 8 RCT/brigades (rather than the 3 to 5 divisions generally quoted). There is probably room for another RCT on both Omaha and Utah if landing craft are available; plus perhaps 1 more brigade on British beaches.
An earlier landing means the beach defences are weaker (both in terms of material and troop numbers), and better weather makes landing easier, leading to deeper penetrations on the first day. In particular a stronger landing on Utah, plus an airborne division landing on the west coast of the Cotentin (as the rearrangement of German deployments which prevented it will not yet have happened) will lead to Cherbourg being opened much earlier.
On another note: How will Monty deal with the "logistics kings"? Ike could not put his foot down.
Montgomery had no great problems with his own logistics structure and supplies, (apart from the elusive 'engine problems') and probably had a greater understanding of logistics issues than most from his experience in North Africa.
A faster initial penetration and earlier opening of Cherbourg reduce the logistics issues in any case, and preventing an Army heading off towards the Saar will help further.
The real issues are the interface between the US Ground Forces and Services of Supply, which needs a strong US theatre commander to take action - which Eisenhower never really did - and an early decision to either capture or mask Brest, or carry out Operation Chastity.
There still would have been logistics issues but they would probably have become important after a Rhine crossing rather than earlier.
Would a month have decided anything? Starting Overlord in May vs June does not seem to give much more room for operations.
IIRC the May weather was OK and the June storm which wrecked Mulberry A would not have been significant, as Cherbourg would be open.
As noted above a May landing also sees weaker beach defences as for example 352 Division would not have moved forward.
The key difference is reaching the position to launch a Rhine crossing in early August, not mid-September; longer days and better weather gives an airborne operation a much better chance of success.
After the Ruhr, the target will be Berlin and I cannot see Montgomery stopping at occupation boundaries. When the Germans would give up I don't know, but I can see further German attempts to assassinate Hitler.
No Anvil of course also probably leads to the Allies reaching the Po in the summer of 1944, but delays the rebuilding of French Armies in France.