I've always allowed a mobcap / hair net to go over a hair covering as long as it covers the covering as much as it would without and doesn't compromise things like hair being covered.
I don't see the difference between a woolly hat and hair as long as it's covered and I don't see then need (nor how you'd police) a mobcap / hair net underneath.
In the UK is it's not just warm head gear but hijabs and turbans which also need to be covered and you're going to be getting into an ethical minefield if you insist either need to be removed to put on a hair net or mob cap first.
With a woolly hat or stocking hat, then the hair net needs to cover it. Same with a turban.
With a hijab, that's generally not possible. I ask for no pins in the hijab and for it to cover the head area including where the ears would be uncovered. I do allow the rest of the hijab to cover the neck as long as the bottom of it is covered by the coat. The reasoning being any hair caught under the hijab is not likely to fall into the product and you don't wash your neck before walking into production so I don't see the difference. The advantage of doing it that way is you don't then need to insist on a company issued and laundered hijab and have risk about it being put on, hair on the outside etc but also if there are any grips / bobby pins, they'll normally be covered by the net.