If the arrest was recent
Start with the county jail
In New York, someone who was just arrested and not released is held in a LOCAL jail, not a state prison, so the DOCCS Incarcerated Lookup will not show a brand-new arrest.
In New York City (the Bronx, Brooklyn, Manhattan, Queens, and Staten Island) that means a New York City Department of Correction (NYC DOC) facility such as Rikers Island; use the NYC DOC Inmate Lookup Service (Person In Custody / P.I.C.
Lookup) and search by first and last name.
Outside NYC, the person is in the COUNTY JAIL run by the county sheriff, and some sheriffs post an online jail roster.
New York also participates in the national VINE system (New York VINE), a free way to check custody status and register for alerts (search at vinelink.com).
For a very recent booking or bail details, call the facility's booking or records line.
In New York City, use the NYC DOC Inmate Lookup Service (a073-ils-web.nyc.gov/inmatelookup). In the other 57 counties, jails are run by the county sheriff and some post an online roster. Use New York VINE or the sheriff's roster, and call the jail's booking/records line for recent bookings and bail amounts.