Another phrase for "jumping on the bandwagon"

Powerthesaurus has a number of alternatives.

I think that 'follow suit' is a close synonym, in the sense of cardplayers following the lead of the first player and playing a card of the same suit.

to do the same thing:

When one airline reduces its ticket prices, the rest usually follow suit.

Cambridge

Rakuten claims huge edge cloud, as other operators follow suit

Rethink Research - 22nd November 2019

After Barbara was accused by a fellow student for being racist, Sara also piled on.

pile on.
US, informal.
: to join other people in criticizing something or someone in usually an unfair way.
After the first few negative reviews, all the other critics started piling on.
— MW