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We combine the isothermal Jeans model and the model of adiabatic halo contraction into a simple semi-analytic procedure for computing the density profile of self-interacting dark-matter (SIDM) haloes with the gravitational influence from the inhabitant galaxies. We show that the model agrees well with cosmological SIDM simulations over the entire core-forming stage and up to the onset of gravothermal core-collapse. Using this model, we show that the halo response to baryons is more diverse in SIDM than in CDM and depends sensitively on galaxy size, a desirable link in the context of the structural diversity of bright dwarf galaxies. The fast speed of the method facilitates analyses that would be challenging for numerical simulations -- notably, 1) we quantify the SIDM halo response as functions of the baryonic properties, on a fine mesh grid spanned by the baryon-to-total-mass ratio, $M_{\rm b}/M_{\rm vir}$, and galaxy compactness, $r_{1/2}/R_{\rm vir}$; 2) we show with high statistical precision that for typical Milky-Way-like systems, the SIDM profiles are similar to their CDM counterparts; and 3) we delineate the regime of gravothermal core-collapse in the $M_{\rm b}/M_{\rm vir}$-$r_{1/2}/R_{\rm vir}$ space, for a given cross section and a given halo concentration. Finally, we compare the isothermal Jeans model with the more sophisticated gravothermal fluid model, and show that the former yields faster core formation and agrees better with cosmological simulations. We attribute the difference to whether the target CDM halo is used as a boundary condition or as the initial condition for the gravothermal evolution, and thus comment on possible future improvements of the fluid model. We have made our programs for the model publicly available at https://github.com/JiangFangzhou/SIDM.