Hospice care at home for my dad?

My dad is at home and needs 24/7 hospice care. He has end-state dementia and severe dysphasia. He also is in terrible pain. He is in New York City and I live in California. I have two sisters, one sis resides close by and the other in the suburbs.

He has caregivers through a home care agency around the clock now - they do 12 hour shifts throughout the week. This is the same agency that we have used for caregivers over the past several years for day shifts. Caregivers are not supposed to go shopping or do errands - I do all the online ordering for food and household items.

I am especially worried that he has had two falls just in the last week on their watch. In the last fall (yesterday), he actually hit his head, was bleeding, and the aide called EMS (though she is supposed to call the hospice number). She was present when it happened; I do admit my dad is a jumper and seems to get up quickly - I still feel that these falls shouldn't be happening on their watch.

Also concerning to us is that the aides cannot administer hospice medication on an on needed basis to alleviate his pain. Only family or hospice nurse on duty, if available to come over, can give him the pain meds. Sometimes this also makes me wonder if he is better off in a facility.

What is the best and fastest way to find private care in these circumstances? I think an agency that screens and recommends caregivers might help streamline the process. Thanks!