I don't have family nearby, so I have that problem as well.
I'm blessed to have an older friend who lives alone (near my parents age) who i can call on occasionally in these situations & not feel like i'm spreading germs to a whole family or daycare community. She adores my kids and has "been there" herself & doesn't get upset over things that some others sitters would, like a runny nose or fever. Age does give greater perspective! Perhaps, if you don't have family near enough to help in these situations, you could cultiivate a similar surrogate-grandparent relationship for your child, either with someone in your neighborhood or faith community, it would benefit all concerned.
Good luck!