It's unclear, but seems unlikely.
Consider how the Gamgees are introduced (emphasis mine):
No one had a more attentive audience than old Ham Gamgee, commonly known as the Gaffer. He held forth at The Ivy Bush, a small inn on the Bywater road; and he spoke with some authority, for he had tended the garden at Bag End for forty years, and had helped old Holman in the same job before that. Now that he was himself growing old and stiff in the joints, the job was mainly carried on by his youngest son, Sam Gamgee. Both father and son were on very friendly terms with Bilbo and Frodo. They lived on the Hill itself, in Number 3 Bagshot Row just below Bag End.
Fellowship of the Ring Book I Chapter 1: "A Long-expected Party"
If the Gamgees tended other gardens, it seems odd that they be introduced with no mention of that fact. Then again, there's a narrative logic in focusing on the Bagginses to the exclusion of all else; Bilbo is the character that contemporary readers would have been familiar with, and Frodo becomes slightly important to the rest of the story.
The biggest reason I think not, though, is non-diegetic. Tolkien made no secret of the fact that he modelled the relationship between Frodo and Sam on that of a young officer and his batman:
My 'Samwise' is indeed (as you note) largely a reflexion of the English soldier—grafted on the village-boys of early days, the memory of the privates and my batmen that I knew in the 1914 War, and recognized as so far superior to myself.
Letter to H. Cotton Minchin, April 1956
Military history is not one of my areas of expertise, so I'm welcome to being corrected on this point, but from my research it seems as though batmen in World War I were assigned exclusively to one officer; the notion of several junior officers pooling the services of a single batman seemed to have been introduced in World War II. From Wikipedia:
The official term used by the British Army in the First World War was "soldier-servant". Every officer was assigned a servant, usually chosen by the officer from among his men. The term batman replaced this in the inter-war years. By the Second World War, only senior officers of the army and Royal Air Force were officially assigned batmen, with junior officers usually sharing the services of one batman between several officers.
The dynamic between Frodo and Sam also seems closer to that of a domestic servant, rather than an employee. Indeed it's fairly unusual to take your contractor on a year-long walking tour of Hell and, as Matt Gutting points out in comments, to give them your house.