Currently visits are mandatory when a mother is 24 weeks pregnant, in the first two weeks of their child's birth then at six to eight weeks from birth. Another check is required when their child is between nine to 12 months old and then finally at two years old.
But the Institute of Health Visiting (IHV) wants to see this boosted so that there are further checks at three to five weeks, three to four months and between the ages of three and five.
This final check should have a specific focus on "school readiness", says the report Health Visiting in England: A Vision for the Future.
Health visitors are able to help spot speech and language as well as communication issues before a child starts school, "including signposting to specialist support if needed", says the report.
Young mothers are a particular group at risk of health problems and becoming disadvantaged, says the report.
- Analysis: Making the case for health visiting
- Analysis:Blueprint for better health and social care integration
It says that in 2016 babies born to mothers under the age of 20 had a 24 per cent higher rate of stillbirth and a 56 per cent higher rate of infant mortality.
"Young mothers are less likely to complete education and may be further economically disadvantaged by a failure to enter employment," states the report.
"Younger mothers are also more likely to smoke during pregnancy than older mothers."
It adds: "Young people who are looked after are three times more likely to be a parent by 18."
IHV executive director Dr Cheryll Adams said the extra visits are also needed to ensure health visitors can better support families to tackle a raft of mental and physical issues, including dealing with bereavement and health problems such as asthma.
"Despite overall improvements in child health, England lags behind other countries on many key health outcomes: infant mortality reductions have stalled, our breastfeeding and obesity rates are amongst the worst in Europe, our immunisation rates are falling and health inequalities are seen across all indicators," said Adams.
"The current status of health visiting is not serving families well, based as it is on universally delivered process outcomes which risk 'ticking the box, but missing the point'.
She added: "Based on the evidence, we recommend that the universal offer in England includes three additional service 'review points', increasing the offer to eight contacts, with additional tailored support where needed."
Alana Ryan, senior policy officer at the NSPCC, added: "While government is currently consulting on a new public health strategy and has committed to updating the Healthy Child Programme, families across the country need to see a bold vision for improvement. This must come soon because without it we will continue to see a service that is variable, with some areas not able to provide a consistent, high-quality face-to-face service."
In June the IHV warned that council funding cuts are leading to an erosion of health visiting services in many parts of England.
It said members from across the country had contacted them with concerns about cuts.
At the same time it emerged that Suffolk County Council was considering cutting health visitor numbers by a quarter.
Last month health visitors in Lincolnshire took industrial action over concerns over cuts to their service, and are due to ballot members on further strikes.