After traveling to Seattle on the bus quite a few times, we suddenly realized that you don't necessarily have to make the connection between the Olympia bus and the Seattle bus at the Lakewood 512 Park and Ride. The 603, 605, and 612 all keep going and stop at 10th and Commerce, in downtown Tacoma, by the State Historical Museum and across from the entrance to the UW Tacoma's campus.
By walking across the street there, you can catch the Sound Transit 594 to Seattle there instead of at the Park and Ride. This has two advantages. If you are lucky, you can sometimes catch an earlier run of the 594 (the one that that leaves the Park and Ride shortly before your Olympia bus is scheduled to arrive there. Sometimes it's a little late leaving the Park and Ride, and it always takes longer to get from the 512 to 10th and Commerce than the Olympia bus does because it makes an extra stop in South Tacoma in between that the 605 doesn't, so you sometimes get there before it does.) And if you aren't lucky, and you still have to wait and catch the 594 that you would have been waiting for anyway at the Park and Ride at this stop downtown instead, you do have several cafés in which you can wait for it that are considerably nicer than standing in the wind at the 512.
To illustrate, we usually catch the 605 that leaves Olympia at 10:30. It's scheduled to get to the Park and Ride at 11:20. Thus, it's scheduled to miss the 594, which was supposed to leave the Park and Ride for Seattle at 11:08, twelve minutes before you got there. If you get off at the Park and Ride and wait for the next 594 there, it's scheduled to arrive at 11:38, so you have an 18 minute layover. However, if you just sit tight and stay on your 605, that's scheduled to get to 10th and Commerce at 11:40. Lo and behold, the 594 that left the Park and Ride before you got there isn't scheduled to get to 10th and Commerce until 11:33, only seven minutes before you get there. Sometimes it's a little late, and/or you're a little early, and you can hustle across the street and get right on it. (This happens between a third and half the time, in our limited experience.) If you aren't lucky, the same 594 that you would have been waiting to catch in Lakewood at 11:38 arrives to pick you up downtown at 12:03. (You now have had a half hour layover, but you've been sitting in the Renaissance Café sipping a delicious milkshake instead of standing around stamping your feet in the cold at the Park and Ride.)
This particular example only works on the morning buses, which skip the Tacoma Dome stop completely. In the afternoon, I think that staying on until you get to the Dome is even more promising. The 12:00 bus from Olympia gets to the Park and Ride at 12:45, missing the preceding 594 by seven minutes, but it gets to the Dome station at 1:00, three minutes before that bus is supposed to get there. If you get off at the Park and Ride you wait 23 minutes for the next 594, at 1:08.
Your mileage may vary...! My main advice is that studying the schedule a little more carefully might be useful (especially since the integrated Seattle to Olympia schedule that Intercity Transit now has on their website just tells you to transfer from Olympia to Seattle at the Park and Ride.)