The Royal Voluntary Service
Do you not have anything like this over there...?
The RVS is a part of the National Health Service (NHS) and has a number of roles, all carried out by volunteers, including visiting those patients who may be alone or otherwise have nobody to visit them.
When my father in law was in hospital for his triple by-pass operation last year, one of the volunteers came in to sit with an elderly gentleman in the bed opposite and she simply sat and talked with him. She came back every day that he was in hospital and said she would visit him at home after he left.
She stayed for a couple of hours and it not only made his day, but he said after she'd gone that she'd actually given him the will to live again as he had felt depressed and alone for a long time.
Being in hospital or a nursing home can be a sad and lonely time for those who have no living family or relatives. Sure, the medical staff are caring and will cater to a patient's physical and medical needs, but they're usually rushed off their feet and just don't have the time to sit for any length of time to really give somebody the sort of attention that would make such a big difference to them.
I would be very surprised if there is no equivalent to this service in the US. But if there isn't, then why not start a volunteers group in your area, publicise it online and try to get it started...?