May was the first flight for it, but I would look for a service entry date or a date of first action. Spitfire: The History didn't provide that in detail. It does support that at least by October they were in active service, though only 50 had been built.
Keep in mind, not that many Seafire Mk II were built, only 200-250 as I recall, before the Seafire Mk III with folding wings came into production, of which more than 1200 were built.