I found hospice was a great help and enabled a lot of services once my parents started it. It made the dying process a lot easier for each of my parents when the time came and for me as their caregiver. Some things you cannot prevent no matter how much you want to, and having that service was a comfort and huge assistance.
Hospice varies by hospital. Make sure the hospital and their hospice meets your needs/expectations.
Ask a lot of questions about what you can expect and how often hospice will come in to the home before the patient leaves the hospital (assuming the hospice is at home).
Again, if done at home, make sure the intake is done the same day, as soon after the patient comes home as possible.
Talk to them about the patient's ongoing medications and how/whether they'll be continued at home.
Look into hospital at home programs to see if that would be an option. It is my understanding that far more in care is provided in these programs.
If I think of more, I will post again. We unfortunately had a rough experience with hospice earlier this year which has made me think a lot about what we could have done differently to make it go smoother.
ETA: If at home, have an understanding of how much lifting and things of that nature will be needed, and think realistically about whether the caregiver(s) will be able to physically handle it.
Just to note, in the US many hospice programs are not directly under just one particular hospital. They may work with hospitals and physicians in an area, but the hospice itself is often an independent provider.