Hampshire's racing career began before the Second World War. He first appeared at the 1939 Nuffield Trophy at Donington Park, racing alongside established names including Bira, Luigi Villoresi, Tony Rolt, Raymond Mays, Peter Whitehead, Leslie Brooke, and Reg Parnell. The Maserati 6CL he drove — formerly owned by Arthur Dobson and modified to 1,100 cc — retired early with a melted piston; the car was subsequently returned to its original 1,500 cc configuration. Hampshire took the 6CL out again at the 1939 Brooklands Whit Monday meeting and at the Sydenham Trophy at Crystal Palace on 20 May, a race televised by the BBC. A final pre-war outing came at Donington Park on 12 August 1939, just weeks before the outbreak of World War Two.
With British circuits largely unavailable in the immediate aftermath of the war, Hampshire first returned to competition through sprints at Gransden Lodge and Shelsley Walsh. In 1946 he ventured overseas, driving his Delage 158L — a 1927 Grand Prix car — at the Albi Grand Prix and the Grand Prix des Nations, where he finished eighth.
The 1947 season brought two outings: he first drove Reg Parnell's car known as "The Challenger" in the British Empire Trophy, then handled Parnell's ERA E-type at Lausanne, both ending in retirement.
Hampshire's 1948 campaign proved more rewarding. He took second place in his Delage 158L at the British Empire Trophy on the Isle of Man at Douglas and finished seventh at the Zandvoort Grand Prix in the Netherlands. He also entered the Jersey Road Race that year.
For 1949 he acquired a ten-year-old ERA R12b. The season brought a fourth-place finish at the British Grand Prix at Silverstone and another fourth at the Goodwood Trophy. He also recorded top-ten results at the Jersey Road Race and the British Empire Trophy, an eleventh position at the International Trophy Meeting, and retirements at the Albi and Lausanne Grands Prix.
During the 1950 Formula One season, Hampshire competed in two World Championship Grands Prix and a number of non-championship Formula One races. A highlight came at the non-championship Jersey Road Race, where he took pole position in the Scuderia Ambrosiana Maserati 4CLT; a magneto failure prevented what would have been his first Formula One win, though he set the fastest lap of the race, which was eventually won by Peter Whitehead in a Ferrari. Later that season Hampshire did claim a Formula One victory, winning the 1950 Nottingham Trophy at Gamston in the same Scuderia Ambrosiana Maserati 4CLT.
In 1951, Hampshire competed in the 24 Hours of Le Mans as part of the works Aston Martin DB2 effort. Paired with his friend Reg Parnell, he finished third in class and seventh overall, a strong result for the British manufacturer in the French endurance classic.
Hampshire represents an important thread in British motorsport history — a privateer and gentleman racer whose career bridged the final pre-war seasons and the earliest years of the World Championship. His pole position and fastest lap at the Jersey Road Race in 1950, even without a win, demonstrated genuine single-seater pace. His Le Mans result with Aston Martin underscored the competitive standing of British machinery in the early 1950s.