If the arrest was recent
Start with the county jail
In Washington, someone who was just arrested and not released is held in a COUNTY or CITY JAIL run by the county sheriff or city police, not a state prison, so the DOC Incarcerated Data Search will not show a brand-new arrest.
Washington participates in the national VINE system (Washington VINE), a free way to check custody status and register for alerts by phone, email, or text (search at vinelink.com or use the VINELink app).
Many Washington county sheriffs also post an online jail register / 'who's in jail' roster (for example King, Pierce, Snohomish, and Spokane counties).
For a very recent booking or bail details, call the jail's booking or records line.
County and city jails are run by the county sheriff or city police in Washington's 39 counties; many post an online jail register. Use the jail's roster or Washington VINE, and call the jail's booking/records line for recent bookings and bail amounts.