June = King Salmon. Saltwater only.
Possibly some Coho (saltwater) by that time but not in any large numbers, certainly not enough for flyfishing. Any you catch will be incidental while fishing King Salmon at that time.
Forget about summer-run steelhead because they simply do not exist. Rainbows are quite few. No sockeyes on the road system (all 25± miles of it). Too early for dogs. Maybe a few dollies. Don't expect much in the streams. I think there may be some Dolly fishing at the stream mouths.
Your best bet is to go mooching for Kings. You'll be there at a prime time for that and the fish will be in prime condition. It aint flyfishing, but it's great sport nonetheless and somewhat of an art too. Don't bother going on a charter boat where trolling is the technique used. Much better to fish actively with your rod in hand, not passively with it propped in some rod holder.