Ten days of silence after a COVID diagnosis warrants a direct inquiry to the facility, and being listed as the emergency contact gives you standing to ask for information.
Call the facility and ask to speak with either the chaplain or the unit team secretary. When you reach them, request what is called a humanitarian update on your inmate. That is the standard term for a welfare check inquiry from a concerned family member or emergency contact, and staff are generally responsive to these requests when they are made respectfully and with the right framing.
The chaplain in particular is often the most accessible and most compassionate point of contact for exactly this type of situation. They handle welfare inquiries regularly and can typically confirm whether an inmate is at the facility, whether they have been hospitalized, and in general terms how they are doing without violating privacy restrictions.
The unit team secretary is the administrative hub for the housing unit and can pull up basic status information quickly. Either contact is a good starting point.
As the listed emergency contact, you have more standing than a general caller to request this kind of information. Identify yourself as the emergency contact when you call and make that clear early in the conversation. That designation exists specifically for situations like this and the facility should be responsive to your inquiry.
If the facility tells you your inmate was transferred to an outside hospital, ask which one and then contact that hospital's patient services directly to get a condition update.