If the arrest was recent
Start with the county jail
In Ohio, someone who was just arrested and not released is held in the COUNTY JAIL run by the county sheriff (or a city/municipal jail), not a state prison, so the DRC Offender Search will not show a brand-new arrest.
Ohio participates in the national VINE system (Ohio 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 Ohio county sheriffs also post an online jail roster / 'current inmates' lookup.
For a very recent booking, bail/bond amount, or the surety and cash-bond options for that county, call the county jail's booking or records line.
County jails are run by the sheriff in each of Ohio's 88 counties, and larger cities also run municipal jails; many post an online jail roster. Use the county sheriff's roster or Ohio VINE, and call the jail's booking/records line for recent bookings and the bail/bond amount.