If the arrest was recent
Start with the county jail
In Texas, someone who was just arrested and not released is held in the COUNTY JAIL run by the county sheriff (or a city jail), not a state prison, so the TDCJ Inmate Search will not show a brand-new arrest.
Texas has 254 counties and jail rosters are fragmented across them.
Texas participates in the national VINE system (Texas 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 large-county sheriffs also post an online jail roster / 'who's in jail' lookup (for example Harris, Dallas, Bexar, Tarrant, and Travis counties).
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 Texas's 254 counties; many large counties post an online jail roster. Use the county sheriff's roster or Texas VINE, and call the jail's booking/records line for recent bookings and bail amounts.