If the arrest was recent
Start with the county jail
In Oklahoma, someone who was just arrested and not released is held in the COUNTY JAIL run by the county sheriff (or a city/tribal jail), not a state prison, so the OK Offender Lookup will not show a brand-new arrest.
Oklahoma participates in the national VINE system (Oklahoma 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); the OK Offender Lookup page itself links to VINELink.
Many Oklahoma county sheriffs also post an online jail roster / inmate lookup.
For a very recent booking or bail details, call the county jail's booking or records line.
County jails are run by the sheriff in each of Oklahoma's 77 counties; many post an online jail roster. Use the county sheriff's roster or Oklahoma VINE, and call the jail's booking/records line for recent bookings and bail amounts.