Very impressive collection James. The cover looks beautiful to me. I would be proud to have it my collection. Along with pretty much everything else you have for that matter.
I have a field grey one with hooks, although I'm very suspicious that it might not be good. I've never had the opportunity to examine an authentic one. The main types mentioned in Baer's books (not including the field made hessian cloth ones) are the winter white camo versions which are well documented as being made in large quantities, and were fitted to the helmet with hooks, and field grey (or just grey) ones, which were not approved by the high command.
Because helmet glare and comouflage was such a big concern for frontline troops, I think many units, especially assault troops, were either issued with covers or had them made up. I've noticed that many wartime pictures show units collectively wearing covers, usually tan like yours, which suggests they were made in large quantities. The same is true with painted camo helmets later in the war in terms of uniformity. Some pictures even show the same pattern of camo on all of the helmets worn in a particular unit.
Here's a picture I own showing covers being worn. These don't have the leather reinforcements, and look like they may not even have holes for the lugs.
The close-up shows that the covers are fastened with drawstrings.
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Detail of Pioniere from the 2nd Company M.W. Zug of Sturmbataillon 16. Most are wearing camo helmets identical in pattern, with only the colour panels switched.