There do not appear to currently be any breeding programs specifically tailored to increasing the population of wild chinchillas. There are currently protected areas designed to protect wild chinchillas from people and predators so that their numbers might increase naturally.
I think that this would be a good approach to chinchilla conservation if they have enough in their wild population to rebound.
I think a breeding program from domestic chinchillas to repopulate the wild ones would be possible but better for a last resort if the gene pool of the wild ones disappears. The chinchilla hasn't been domesticated so long as to make it impossible to reintroduce them to the wild. Animals like dogs have been in breeding programs for thousands of years. I do think they would have to study what traits help the survival of a wild chinchilla. For example probably the densest fur would be a disadvantage because of natural moisture and possible overheating. The size would probably be better off on the small side for calorie reasons.
The chinchillas chosen if such a program were made would have to be thought out well. You wouldn't want the show champion, rather you would probably be looking for as diverse characteristics in the animals you pick. Getting some with thin, medium and dense fur, varying lengths, light, medium and dark phases, small size (wild ones are smaller for a reason, harder to get food, and physical fitness) but good hip passage for young. Things that you would not consider for a domestic chinchilla but can still find in the gene pool.
I have thought about this before, usually when dreaming about winning a Lotto jackpot. =P You would definitely have to have phases in the program. The first phase would have to be studying and tagging wild chinchillas. Learning what they need to survive predators and harsh years. Then the second phase you might begin collecting as wide a range of chinchillas that exhibit as many of the needed characteristics as possible while trying to avoid strains with heart, teeth, or breeding issues.
The first set of animals couldn't be dumped in the wild. So you would have to design some type of halfway house so to speak. Maybe an open enclosure, like they have for some of the zoo animals. Have as natural habitat as possible with the exception of providing safety from predators, food and water. Raising the chinchillas in breeding cages would be counter productive to the young. Any chinchilla that you had to treat medically, like hand feeding young, or a troubled birth you could still care for but also have the control of removing it from the population of that program.
The young would be more naturally adapted to the wild environment and probably moved to a second enclosure that more closely resembles the natural environment with even less human intervention (natural water source, more natural plants for forging, ect). I think something like this would work in the long term to get a domestic strain more adapted to the wild, it would also allow the start of natural selection to promote the chinchillas that would ultimately survive better in the wild.
Ok I'll keep the rest of my imagination in check, but if I do ever win 300 million I'll be sure to get professional conservationists to help with the project. ;D
Edit: PS while genetic testing would be the biggest benefit, good old fashioned pedigree sifting and selection by phenotype would still be beneficial. The key would be getting chinchillas with a wide range in the traits so that the gene pool would be the largest possible and natural selection could take over.