The move would shrink a long-standing productivity gap with countries such as Germany, Italy and France, Greengauge 21 said.
It would also reduce an over-reliance on London by lowering journey times and allowing other cities to flourish, it added.
A new connection in the West Midlands would see HS2 trains speed to cities such as Bristol, Cardiff, Newcastle and Edinburgh, reducing the need for environmentally harmful cars, the group suggested.
HS2 is a planned high-speed rail network, with trains holding more than 1,000 commuters travelling up to 250 mph, from London to Birmingham and to Manchester and Leeds.
Services on the London-Birmingham phase of the £56 billion network are predicted to open from 2026.
Greengauge 21 director Jim Steer said: “Britain lacks a long-term national railway strategy beyond HS2.
“We need a plan to put rocket fuel into our economic productivity and today’s report sets out proposals to do so.
“It is vital for the future of the country that no region is left behind, and the national railway strategy needs to reach all parts of the country.”
The report, titled Beyond HS2, said boosting national productivity should be the guiding priority for re-designing the rail network.
Key proposals include:
1. An upgraded fast route from Birmingham to Bristol Parkway carrying HS2 trains, continuing to the South West and South Wales.
2. A major upgrade to the East Coast mainline for the first time since the 1980s.
3. New high speed lines in Scotland, achieving a three-hour 15-minute journey time between Edinburgh and London.
4. New lines in East Essex and Anglia, alleviating the West Anglia and Great Eastern Mainlines, both of which are at capacity.
5. Bringing Liverpool, Manchester, Leeds, Bradford, Sheffield, Newcastle and Hull together with new connections to form “an effective and powerful economic unit”.
Mr Steer added: “Fundamentally, we need to completely re-orientate the railway from a ‘hub-and-spoke’ centred on London to a fully national network.”
Workers employed in the UK have lower productivity levels than their continental counterparts, the group said.
It is 10.5 per cent lower than Italy, 22.8 per cent lower than France and 26.2 per cent lower than Germany, it said, based on a current price GDP per hour worked basis.
A DfT spokeswoman said: “This Government has an ambitious and clear strategy for the future of our rail network.
“HS2 will provide the backbone of our railway system, improving connections between our major cities, boosting productivity, delivering better journeys for passengers and driving economic growth across the country.
“With Crossrail nearing completion and plans progressing on Northern Powerhouse Rail, we are ensuring the whole country has the connections it needs.
“Whilst we may not agree with all recommendations in this report, it is an important contribution to the debate and underlines the need for HS2, delivering the rail network this country needs for the future.”