In times of urgency as noted both sides made use of whatever was at hand to move wounded off the fields. I always caution folks that find and use various adverse condition references, since human nature is to write of the unusual or lack of resources available at a given time... v/s what was normally a boring daily routine habit. A soldier might write after an engagement they had no ambulances available... That on the surface may imply none existed... Then further detailed research may uncover that the army had 200 ambulances with them but they had gotten stuck and bogged down in the muddy roads 10 miles from the battlefield....
As in the ones known to have been made in Richmond, it is most likely there were many other such contracted makers in other large Southern cities likewise that were linked to a QM depot system of sorts. That could have been a supply base for other theaters. The Ambulance itself was not a complex vehicle that could only be made in specialized military shops. Just about any carriage & wagon maker (nearly every local town usually had several) could make these. The omnibus frequently found in the larger cities were basicly an ancestor to the passenger shuttle bus (different from a stage coach). Local hires or owned by local businesses in the 1850-60's era. Intended to carry passengers to and from railroad stations to hotels etc around the immediate area. The basic spring chassis design on many of these omnibuses as well as some carriages were quite similar to that utilized on many of the ambulances. So it would not be unusual or difficult for the local QM depot to contract to many of these local carriage makers to make Ambulances as well... We have found lots of evidence of this local/regional private contracting habit existing with many other necessary items and goods in use by the military.
If your looking for specific confirmed evidence of use in a given area, you may wish to research what troops or forces were there.... then what supply depots supplied those troops. May take a bit of extra research time but survey of those related Quartermaster contracts, vouchers, invoices, and issue returns. (The Quartermaster Dept usually were the ones that contracted and supplied such items to a given army force) This frequently will turn up additional secondary evidence and clues what may have been commonly in use and in what numbers. The local assigned Medical Dept returns and reports might also give reference to what they might have had on hand at a given time frame.
Respects
Frederick