I'm always happy to learn more - I haven't come across the Tees find, so I'd be interested in any details (including any dating - though I appreciate it might be difficult if it's from a river).
The evidence from depictions (manuscripts and coins) does point rather strongly towards conical helmets being very common (I was going to say "normal", but I won't), on the assumption that heads weren't triangular. Whether the 'coxcomb' headgear represents a metal helmet, a non-metal helmet (leather?) or a civvies Phrygian cap (potentially felted) doesn't seem to have any consensus.
Domed helmets are well attested earlier on (seventh to ninth/tenth centuries), though the fashion seems to have been changing in the tenth century; by the eleventh conical is the norm for headgear-that-is-unquestionably-a-helmet. The ‘Gjermundbu’ helmet (buried mid-tenth century) fits in the tradition of the sixth and seventh-century Vendel/Valsgard helmets, which were highly decorated and found in royal graves.
Given the longevity of military kit and the value on metalwork of any sort I wouldn't want to rule out much earlier helmets still being in use. However, the armies of 1066 were composed of well equipped semi-professional and professional aristocratic warriors able to afford good equipment. In this context I'd be surprised to find a thegn, miles or hirdman wearing a helmet inherited from grandfather's grandfather.
On the basis of the available evidence (and the fact that spectacle helms are, unusually, something that will stand out from a distance), we will and do take a fairly strong period-specific view. If you're reading this and are coming to Hastings, you'll know in advance and can hopefully bring kit accordingly, so we'll look forward to seeing you.