When someone is in the hospital, if we find out about it we usually make a note in F1 and then assign a pastor to visit. After the initial visit, we (obviously on a case by case basis) pass it on.
We have a congregational care team to pass it on to.
The lead of the team owns a business and is always available. They mobilize their team and send cards, flowers, and do follow up on the person in the hospital.
Ideally, this team reports to the care team pastor weekly and the information is updated in F1 notes.
It isn't a fail-proof method, but it seems to be working so far.