  We combine the methods of Hamiltonian Truncation and the recently proposed
generalisation of the S-matrix bootstrap that includes local operators to
determine the two-particle scattering amplitude and the two-particle form
factor of the stress tensor at $s>0$ in the 2d $\phi^4$ theory. We use the form
factor of the stress tensor at $s\le 0$ and its spectral density computed using
Lightcone Conformal Truncation (LCT), and inject them into the generalized
S-matrix bootstrap set-up. The obtained results for the scattering amplitude
and the form factor are fully reliable only in the elastic regime. We
independently construct the "pure" S-matrix bootstrap bounds (bootstrap without
including matrix elements of local operators), and find that the sinh-Gordon
model and its analytic continuation the "staircase model" saturate these
bounds. Surprisingly, the $\phi^4$ two-particle scattering amplitude also very
nearly saturates these bounds, and moreover is extremely close to that of the
sinh-Gordon/staircase model.
