I cannot see a problem feeding spinach during moult when birds are not actively laying, since the calcium absorption during that dormant phase is not important and maybe the spinach even makes the layer feed better for them in that respect by preventing them from absorbing that excess calcium that they don't need.
Interestingly I also found my girls went mad for spinach during their moult this autumn and would hardly eat any of their regular layer feed. I swapped to a grower but they were equally disinterested. Poops were very runny due to their largely green vegetable diet and I did increase their scratch at that stage as I was concerned that they were seriously losing weight.... I pick them up almost every night because they have a habit of roosting on top of the nest boxes and I can't be bothered to change the lay out of the coop, so I could feel how lightweight they had become. I probably only increased the scratch for a week or so and maybe doubled the small amount I normally give them. They eventually started eating their formulated feed again when they were ready and nothing changed, so it wasn't that the feed was off or anything.
Personally I feed spinach year round without any problem in shell quality, but if I was to start having problems, that would be the first thing to cut out.