If the arrest was recent
Start with the county jail
In New Mexico, someone who was just arrested and not released is held in the COUNTY JAIL (often called a county detention center), run at the county level, not in a state prison, so the NMCD Offender Search will not show a brand-new arrest.
New Mexico participates in the national VINE system (New Mexico VINE), a free way to check an offender's custody status and register for phone, email, or text alerts (search at vinelink.com or use the VINELink app).
Many New Mexico county detention centers also post an online jail/inmate roster.
For a very recent booking or bail details, call the county detention center's booking or records line.
County jails/detention centers are run at the county level in New Mexico; several larger and metropolitan counties post an online inmate roster. Use the county roster or New Mexico VINE, and call the jail's booking/records line for recent bookings and bail amounts.