Two Jessops stores will open in London's Oxford Street and Birmingham, with four others opening over the Easter weekend in Manchester, Aberdeen, High Wycombe and St Albans.
Up to 30 more Jessops stores are due to reopen in April, as part of a £4 million investment.
Jessops called in administrators PwC in January, leading to the closure of all 187 Jessops stores and the loss of almost 1,500 jobs.
Later that month Mr Jones and other investors bought the brand and certain assets. The entrepreneur is the sole owner of the relaunched camera store chain.
The company has pledged to create 500 jobs with its relaunch - with the majority going to former Jessops staff who lost their jobs when the chain collapsed.
Mr Jones, who appears as an investor on the BBC reality TV series about entrepreneurs, will take the role of chief executive and chairman.
He said: "I believe Jessops is an iconic British brand which can lead the retail resurgence on Britain's high streets, powered by new innovations and world-leading, expert staff. Image is everything, and even in the mobile and tablet era, there's no substitute for a quality camera when it comes to taking the perfect picture."
The relaunched company is forecasting sales of at least £80 million in its first year, which it said will give it 15% of the DSLR camera market in the UK. Jessops also promised greater integration with its online presence, including an option to collect internet orders at store.