As a general FYI because I have a friend dealing with an emergency right now, some airlines offer compassion fares or medical emergency or bereavement fares. If you have a family emergency and you need to fly right away, call the airline directly and ask.
They might require some proof like the name of the hospital/patient/funeral home but if you have to fly right this moment, it’s an option. The airlines might also allow you more flexibility on flight changes as well.
I know this applies for Delta, at least, and even if an airline doesn’t explicitly offer emergency fares, they may still be able to help you out.