The only way to hide it is to use a PO Box as your FCC mailing address. As a FCC licensee your contact information is public for a few reasons, like any other TV or radio station. It's also public for your convenience, so that other hams know where to send QSL cards. I don't know that it's a major problem unless you're telling patients your call sign and even then do you suppose they will they know where to look?
'91 Pickup - Imelda / '08 Tacoma TRD - Donna
'09 Kawasaki KLR650
'98 Rhygin Juke SS / '12 Gunnar Rockhound 29 / '16 REEB Dikyelous
"Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security. " --Albert Einstein