Well hairy is not a term I'd want used to describe my beer! That sounds like a pellicle infection. Images seem to be loading hold on.
Ok, honestly that looks like it might be the early stages of a pellicle to me. Maybe a nice SOUR beer??
And headspace won't cause it. It might cause some oxidation if there isn't enough residual CO2 from the primary to push all of the air out of the headspace. I'd look for anything that may harbor bacteria, like unsanitized hands, mold or old beer in the transfer hoses, spigot (if your primary was actually a bottling bucket), autosiphon, etc. Generally a fermented beer will have enough alcohol to inhibit the growth of low levels of bacteria.