You have a lot of options in the Tacoma area. There are several hospitals in Tacoma and the Tacoma area that have new graduate residencies. CHI Fraciscan has several hospitals in the Tacoma area as well as one in Bremerton. One of the hospitals is even located on Gig Harbor. Their new graduate residency is a bit different than some others in the area because they have a rolling start system; you do not have to wait for a cohort to start. They have classes every month and just add you in when you start working. The exception to that is up in Bremerton; their hospital has a Versant Residency and it starts I believe twice a year. I am just getting ready to start working for them myself this week. If you are interested in any positions; you just apply for it. I did a search with the keyword residency so that I was applying for units I knew were looking for new graduates. I applied back in June, heard from the unit manager in August, interviewed and was offered the position a few days later. I am working at St. Joseph's in Tacoma. Here is a link to their website Nursing | CHI Franciscan
There are also some Multicare hospitals in the Tacoma area including Tacoma General and Allenmore Hospital. They are also located in some other areas. I did not apply with them because I had already accepted a position before they posted their residency positions. Here is their website RN Residency Program | MultiCare Health System
Providence St Peter's hospital is also located down in Olympia and has what seemed to be a good residency program. I did not get selected for the unit that I really wanted so I didn't accept them offer. Providence Nurse Residency Jobs
This website gives limited information about the residency and you will actually need to go to St Peter's website to see when they open up the applications. They do a online video screening interview and then if they like it you get an in person interview. They don't narrow you down to a specific floor but will bring you in and have you interview with multiple managers and then they pick which floor you get an offer on (but they do ask your preference). The major disadvantage I thought with this system is I did not get to tour the units so I didn't get a good vibe on the unit culture and didn't really get a chance to connect with the unit personnel like I did on my interview at St Joseph's. There were 5 different units at my interview.
Honestly, traffic is a real problem in the area. If I lived on Gig Harbor I would definitely want to work in Gig Harbor or in the immediate Tacoma area -- anything to stay off of I-5. I don't know how the commute would be up to Bremerton. You shouldn't have any problem getting a new graduate residency offer in the area and I was able to get a job on my unit of choice. There's lots of options and it's easier to get a position in Tacoma / Olympia because everyone seems to want to try and get up in one of the big name hospitals in Seattle. I did an interview up at Swedish in Seattle and the recruiter told us that they had about 1800 applications for 80 residency spots -- pretty crazy!
The best piece of advice I can give you is to make sure to have a good cover letter and tailor it to the position you want and work on your resume. People who don't put in cover letters typically do not get interviews. My unit manager specifically told me that she decided to interview me based off my resume. Apply before you graduate -- you do not need to have your license. You will get an offer that is contingent on passing the NCLEX. There should be lots of options available for the August cohorts / and CHI positions in the same timeframe. When you start your last semester keep an eye out daily for the residency positions to open their applications because they will open up the application pool and often will stop accepting them with no notice. I would guess that they will start taking applications for the August cohorts around March or even a bit earlier -- just keep checking and you can contact the recruiters and get a estimated time frame when it gets a bit closer.