It should come as no surprise that, as the 40th anniversary of his untimely death approaches in June, Tubby Hayes remains an inspiration to aspiring young musicians and the subject of the sort of complete-set collecting among fans that's usually reserved for American jazz musicians such as Bill Evans and Art Pepper.

The often-maligned CD reissue avalanche that, in pop and rock music, might be considered cashing in on old recordings that have already been paid for many times over, has brought forth dozens of Hayes sessions that were thought lost, previously unknown or just plain unavailable. And as the latest of these releases, Proper Records' 4CD box Little Giant Steps, illustrates, the consistently high quality of Hayes' work as a saxophonist, flautist, vibes player, composer and arranger is deserving of being shouted about from the rooftops.

Hayes wasn't quite an infant jazz prodigy. He began tenor saxophone at the age of 12, having previously taken violin lessons from his studio violinist father and learned to play the piano, and within just a couple of years he had attained a frightening proficiency. We know this from the testimony of Ronnie Scott, Soho jazz club owner and himself no mean tenor talent.

When Scott arrived at a jazz club in Raynes Park in South West London as its star guest one night in 1950, the then 15-year-old Hayes had been playing with the house band for a few months and was persuaded to ask Scott if he might sit in on a few numbers. Scott agreed and later recalled that he'd been joined on stage by "a little boy, not much bigger than his saxophone," who put his instrument to his lips and proceeded to scare him to death. Hayes wasn't exactly a boy but he wasn't tall, hence the Little Giant Steps of this latest Properbox and its predecessor, the Little Giant.

Eight years after their auspicious meeting, the twin tenor partnership of that evening would be fronting one of the fabled bands in British jazz, the Jazz Couriers. By this time Hayes, who had turned professional not long after his joust with Scott, had had big band experience with Vic Lewis and Ambrose, where he learned much about arranging from Edinburgh-born composer Johnny Keating and gained valuable insights into rhythm from his own future drummer, the brilliant but wayward Phil Seamen. Hayes had also led his own band, which he candidly admitted was a disaster.

When Little Giant Steps picks up his story, told superbly in the accompanying booklet by Hayes expert and fellow saxophonist Simon Spillett, Hayes has seen the Jazz Couriers collapse due to the limitations of the British jazz scene of the time but has remained so high in Ronnie Scott's esteem as to be considered the obvious choice to open Scott's jazz club in 1959. He had also recorded with American musicians for the first time – a session for Blue Note Records, no less, with trumpeter Donald Byrd and drummer Art Taylor guesting on Jamaican trumpeter Dizzy Reece's Blues in Trinity album – and had experimented with multi-tracked recording years ahead of the Beatles.

The years covered by Little Giant Steps (1959-1962) were similarly momentous. The set begins with Hayes leading his quartet featuring pianist Terry Shannon, bassist Jeff Clyne and the shortly to be ignominiously sacked Seamen. All the Hayes qualities are present: great tone, huge drive, implacable vigour and exuberant, virtuosic soloing that would lead some to say he played too many notes. But as Hayes would say in his own defence, he loved to play ballads too and was as thoughtful at that tempo as he was uncatchable on the fire-starters such as The Surrey with the Fringe on Top.

Hayes may have felt British jazz players lacking once he had achieved the feat of being accepted in America but as the Properbox illustrates, that didn't stop him leading a home-reared big band that could have held its own with most others of the time.

Little Giant Steps documents his New York recordings, where he more than held his ground with Clark Terry, James Moody, Roland Kirk and Horace Parlan and the quintet he formed with Dundonian trumpeter Jimmy Deuchar. It too was masterly, although it's tempting to speculate if Deuchar had any input behind Hayes's In the Night having an opening sequence that's very similar to Ye Banks and Braes o' Bonnie Doon.

Hayes would go on to work with Charles Mingus and dep for Paul Gonsalves in the Duke Elllington Orchestra. He was also reunited briefly with old pal Victor Feldman in Los Angeles and as a seasoned session player he could easily have joined Feldman in his illustrious studio work. Whether that would have prevented his heart problems or stopped him dying on an operating table aged 38 in 1973, no-one knows. What is certain, though, is that in his short life Hayes created an astonishing legacy. Little Giant Steps is a great place to start and happily, there's more, much more out there.

Little Giant Steps is available through Proper Records.