In the UK a midwife visits the newly discharged mother and baby for up to ten days - even making two visits a day if s/he feels it is warranted.
Usually when the infant is 10-12 days old (but up to one month old if the midwife feels she needs to stay involved) the Health Visitor takes over the care, visiting prn, identifying health care needs and finding ways to meet them, not necessarily doing all the 'work' her/himself. There is such a variety of need for both mother and infant.
In years of health visiting, I found it a very fulfilling job, with every day a little different. The job description is very, very wide, but certainly there is a huge need for the mother and baby part of it in the USA. (Obviously the HV met the mothers prior to delivery.)
RHVs in the UK run their own immunization clinics (with an MD on the premises) and well baby clinics and follow thru right up to pre school visit. They have powers of referral and are trained to do basic tests on hearing and vision which pick up problems earlier than I see done in the USA.
RHV requires RN with at least part of the midwifery course and varied RN experience. It's one year in HV college (sociology, social policy, psych., epidemiology, statistics, HV theory and practice, primary care issues, child abuse, public health etc) with fieldwork practice then about six months in supervised practice. The job can be very different depending on geography e.g. inner city, retirement area, immigrant area etc.
I'm sure an enterprising OB RN could develop a practice for herself in this sort of work in the USA.