The most general renormalizable interaction between the Higgs sector and a new gauge-singlet scalar S is governed by two interaction terms: cubic and quartic. The quartic term is only loosely constrained by invisible Higgs decays and given current experimental limits about 10% of all Higgs bosons at the LHC can be converted to new scalars with masses up to m(Higgs)/2. By including this production channel, one significantly extends the reach of the LHC-based Intensity Frontier experiments. We analyze the sensitivity of the FASER experiment to this model and discuss modest changes in the FASER 2 design that would allow exploring an order-of-magnitude wider part of the Higgs portal's parameter space.