Our Schengen visa to France was refused on the basis of, what I understand now, some basic errors with the documentation provided. We appealed that decision, but in retrospect I do think it unlikely that the appeal will be accepted (failure to provide exact itinerary as it wasn't in required documentation, and a hotel booking lapsing as we didn't make the non refundable payment). I do feel we can make a much better application now. My question is, that as it has been 10 days since we submitted the appeal, how much longer do I wait before filing a fresh application? If I re-apply, would the fact that I have already appealed an earlier decision negatively impact my new application?
According to the French Ministry for Europe and Foreign Affairs
If your visa is refused, you may also submit a new visa application immediately and at any time.
You may appeal against a decision to refuse you a visa by applying to the Commission de recours contre les décisions de refus de visa d’entrée en France (Board of Appeals against decisions to refuse an entry visa to France) within 2 months of the date of notification of the refusal.
As you note, before re-applying, you would want to remedy any issues raised in the refusal, making changes and improvements as appropriate.