Yes, facilities provide winter clothing for inmates during the colder months, and court appearances are covered under that same provision.
Most correctional facilities issue what is generally called a cold-weather kit or winter gear package when temperatures drop. This typically includes a jacket or coat and some form of thermal or long underwear to layer underneath the standard-issued clothing. The quality is institutional and nothing to write home about, but it serves the purpose of keeping people reasonably warm during cold weather movement including transport to and from court.
When an inmate is transported for a court appearance in winter, the transport officers and the facility are responsible for ensuring they are dressed appropriately for the conditions. Standing outside waiting for transport vans in freezing temperatures in a thin uniform is not the standard, and facilities generally ensure the available cold weather gear is accessible when transport is happening.
Beyond what the facility issues, inmates who have funds on their books can purchase supplemental clothing items through the commissary. Long sleeve thermal shirts, heavier socks, and in some facilities additional layering options are available for purchase. For an inmate who goes to court regularly or who runs cold, having commissary funds available to supplement the institutional gear makes a practical difference in comfort.
If you want to make sure your person has what they need going into the winter months, keeping a modest amount on their books so they can shop commissary for any supplemental items is a concrete way to help with something as basic as staying warm.