Hell yeah! They need competition, and I think there's two main sources for that. First is by expanding the Justice Underground; I know that classically, Luthor was the only super-hero on Earth 3, but I'm far more interested in the "Batman: The Brave and the Bold" approach where any villain can be a hero instead. Good versions of Grodd, Cheetah, Manta, Scarecrow, etc.
The other method to create competition for the Syndicate is to have rivalries between super-heroes. Some of this can be infighting between two Syndicate heads who don't want to share territory ("Owlman v. Ultraman: Justice of the Dons" has a nice ring to it, haha!) or it could be between the Syndicate and other super-crime lords who aren't part of the inner circle. I think there's at least one version of Ultraman who claims his first big challenge was a rival criminal Martian?
Though I'll also say, in line with the way that just plain crooks, with no powers, still occasionally give the individual members of the JLA a hard time (tho not the team as a whole), I do think that every so often, a normal cop, detective, CIA agent, should pose a threat to Ultraman, Owlman, Jonny Quick, etc.
Either way, I think there's a lot of room to expand and explore Earth 3, and I think taking it in the direction of just plain old "world domination" is ultimately... kind of boring.