Surprisingly, we know when things started to be mammoth, and that the popularity of the word was sanctioned at the highest and most principled level, with thoughts of publicity, perhaps, but not as part of an advertisement. The occasion was the presentation to President Jefferson of a mammoth cheese made to thank him for his support of religious liberty. Here is one of many accounts: https://constitutioncenter.org/blog/a-tale-of-a-giant-cheese-and-the-first-amendment. The use of the word was clearly inspired by the discovery of mammoth (or supposed mammoth) bones, one of Jefferson’s many passionate interests.
The political associations of the term made it controversial and even derogatory. It did not supplant modest descriptions like “big” and “large.”
But advertisers clearly wanted to wow customers with the prospect of seeing or obtaining something that was more than merely large.
Circuses were often, and naturally, described as mammoth. Among the other advertisements, the only one that has the scale of the mammoth cheese is the mammoth soap displayed at the Centennial. The mammoth shirt collar is delightful, especially as it looks in the picture like a streamlined street lamp. And the owner clearly can’t resist playing with the idea of the mammoth shirt. There is a point in an oversized sign, but would be the point of mammoth products like lamps or (fake) mantels, and would anyone really want a mammoth fashion plate? And children might have nightmares about mammoth asparagus (maybe not about the less aggressive-looking Boston Mammoth White Plume Celery). Maybe the expression succumbed to incongruity.
Finally, thanks to Bob Skiba for mentioning in April 2018 on the Facebook site Vintage Philadelphia the Mammoth Skating Rink and Velocipede Institute of 1868-69. All the fun seen in the notices below ended little more than a week later, when the place was destroyed in a fire.