Every particle has a corresponding field that permeates all of space in the same way the Higgs has a field that does so.
The spin up electron. The spin down electron. The spin up positron. The spin down positron.
The up quarks (all three colors and both spins). The down quark (all three colors and both spins). Same for the charm, strange, top and bottom. And double that because all those quarks each have an antiparticle with the corresponding anticolor and opposite electric charge just like the electron had its antiparticle, the positron.
Then there two more leptons like the electron the muon and the tau lepton (each has two spins and an antiparticle with opposite electric charge).
That's all the fermions that have electric charge. Then there are the eight gluons and they would have three spins each but since they are massless they have two helicity states instead, and they are their own antiparticles)
The gluons are also bosons like the photon, there are only two photon fields, one for each helicity (there would be three spins but the photon is also massless)
There are yet more bosons, the $W^+,$ $W^-,$ and $Z$ each of them have three spins. And the neutrinos are the chargeless fermions and the chargeless leptons. There is one for each of the charged leptons (one for the electron, one for the positron, one for the tao and one for its antiparticle, one for the muon and one for its antiparticle).
Those are the ones we've seen, some people like to predict more. That's already quite a lot and the Higgs is not special (well it is the only spin 0 we've seen so we didn't have to have multiple versions for spin or helicity).
If there is a graviton that would be another.
If someone is talking about early days then I think the idea is the Higgs moves into a lower energy state very early on.