Work for multiple companies?

Assuming you don't have a no compete agreement, or that you are not competing, it shouldnt make a difference. Otherwise, its probably none of their business anyways, but there can be overlaps between caregivers so IDK.

Its worth having a discussion about, especially if you arent competing. 

I won't be case managing so I wonder if that makes a difference.  For my new company it will be SOCs and revisits to help the full timers as its a PT position.  My old job I'd stay on for SOC only. To me the issue is these companies technically do compete in the sense it's the same territory, however there would be no leaving for me as I won't really be following pts. There has been no talk of a non-compete agreement so maybe that won't even come up. But I agree, it's really not their business and nurses hold multiple jobs all the time. 

You visit times must never overlap even being different companies. Would be considered  fraud if audited.  Where I live, there have been fraud cases with billing for more visits than humanly possible.  So, it is monitored. 
If you live in a right to work state, don’t think they can stop you, contract or not.  They can stop giving you assignments.  Getting RNs to do home health start of cares if difficult for most agencies.  A lot of work for the pay.  So, they value nurses who will!  
Many perdiem home health nurses work for multiple agencies.  One gets slow and you have another source for income.  Just never mention to patients or families that you work elsewhere, be careful not to wear the badge of another company when you visit.