Lead halide perovskites undergo pressure-induced phase transition, facilitated by bond compressions and different tilt patterns of the ${\mathrm{PbI}}_{6}$ octahedra. Utilizing first-principles calculations we investigated 14 possible tilt systems of ${\mathrm{CH}}_{3}{\mathrm{NH}}_{3}{\mathrm{PbI}}_{3}$ under compression, derived from the high-symmetry $Pm\overline{3}m$ structure. Fr substitution is adopted to mimic the rotational disorder effect of the ${\mathrm{CH}}_{3}{\mathrm{NH}}_{3}$ cation with the similar size (at room temperature or higher). Analyses of these phases reveal an interplay between tilting and distortion of the ${\mathrm{PbI}}_{6}$ octahedra. Dynamical fluctuations at finite temperature provide additional insight into the phases' stability. Drawing from the trends observed in $\mathrm{Fr}\mathrm{Pb}{\mathrm{I}}_{3}$ and additional calculations for perovskites involving differently ordered organic cations, we propose that phase transition in ${\mathrm{CH}}_{3}{\mathrm{NH}}_{3}{\mathrm{PbI}}_{3}$ occurs from tetragonal at ambient pressure to orthorhombic under high pressure via the pathway I4/mcm$\ensuremath{\rightarrow}$P4/mbm$\ensuremath{\rightarrow}$Imm2. The distinct discontinuity in the calculated volume-pressure curve of I4/mcm and band-gap evolution of the proposed phases are consistent with photoluminescence shifts observed in ${\mathrm{CH}}_{3}{\mathrm{NH}}_{3}{\mathrm{PbI}}_{3}$ under pressure.