The band played Las Vegas and elsewhere in the U.S. It travelled to Finland. It was famous among aficionados in Los Angeles and drew huge audiences there in the 1980s.
Basso, no stranger to the L.A. scene, remembers being there with the Boss Brass when luminaries such as Woody Herman, Johnnie Mandel and Artie Shaw would show up to hear them play to packed houses.
"For me to go back to L.A., with a Canadian band, wearing my Canadian jersey, that was a big thrill for me," Basso said. "That gave the band a huge sense of satisfaction."
At one point in the 1980s, McConnell moved his wife and one daughter to Los Angeles, where he took a teaching job at the Dick Grove School of Music.
But Margaret was especially homesick for Canada, so once again McConnell came home. Eventually, they moved north of Toronto, where he continued his work - playing, arranging and teaching workshops. Margaret died in 2004 of complications from diabetes.
"Rob was a very proud Canadian," said Anne Gibson, whom he married in 2006. "He worked in the U.S. and Europe, with extended stays in New York and Los Angeles, but he always felt he belonged here."
By this time, however, things weren't what they used to be.
The big band revival was fading. Tastes were changing, and the economics of the business saw fewer and fewer venues that could accommodate such a large group. The commercial work that had kept many great players afloat financially was drying up as session musicians were replaced by digital sounds.
"Things change," McConnell said in a 25th anniversary tribute to the Boss Brass by the CBC in 1994. "I don't know why, and I certainly liked the way things were before. I miss all the hanging out and I miss the income.
"But we had a really great time for many years, and I was fortunate to be a part of it."
In 1997, he reorganized the band into Rob McConnell's Tentet and played Toronto venues such as the Senator, the Rex and the Montreal Bistro, as well as festivals and some concerts. The group made several recordings with the Justin Time label and won a Juno in 2001.
Even when McConnell began to suffer from ill health, he didn't stop.
In 2008, he brought the Boss Brass back for a reunion at the Old Mill, prompting The Globe and Mail's J.D. Considine to say: "Perhaps the most dazzling moment was McConnell's imaginative treatment of the standard All the Things You Are, which opened with chords dense and dissonant enough to have been borrowed from Olivier Messiaen, continued with contrapuntal brass lines worthy of a Bach chorale, and boasted a breathtaking piano cadenza by [Don]Thompson. It was a dazzling as big-band writing gets."
The Boss Brass played together the last time on Canada Day, 2009, at the Toronto Jazz Festival. McConnell ended the concert with O Canada, which he liked to do, his wife said, because he could always be guaranteed of "a standing O."
McConnell performed in public for the last time at the Brantford Jazz Festival in September, 2009.
He leaves his wife Anne, children Brian, Jennifer and Robin and seven grandchildren.