Assuming the departure is also from SEA then you just pick up your bags in baggage claim and then walk back upstairs to the check-in area and drop them off for the second itinerary. It is all in one building for baggage claim/check in so no need to go outside much less take a taxi.

There's at least one restaurant in the non-secure side of the airport which might come in handy should the second airline not be open to accept your checked bags, a situation that would not be uncommon 6-8 hours out.