The high-spin nuclear structure of Tc isotopes is extended to more neutron-rich regions based on the measurements of prompt \ensuremath{\gamma} rays from the spontaneous fission of $^{252}\mathrm{Cf}$ at the Gammasphere. The high-spin level scheme of $N=67$ neutron-rich $^{110}\mathrm{Tc}$ ($Z=43$) is established for the first time, and that of $^{111}\mathrm{Tc}$ is extended and expanded. The ground band of $^{111}\mathrm{Tc}$ reaches the band-crossing region, and the new observation of the weakly populated $\ensuremath{\alpha}=\ensuremath{-}1/2$ member of the band provides important information on signature splitting. The systematics of band crossings in the isotopic and isotonic chains and a CSM calculation suggest that the band crossing of the ground band of $^{111}\mathrm{Tc}$ is due to alignment of a pair of ${h}_{11/2}$ neutrons. The best fit to signature splitting, branching ratios, and excitations of the ground band of $^{111}\mathrm{Tc}$ by the rigid triaxial rotor plus particle model calculations result in a shape of ${\ensuremath{\varepsilon}}_{2}=0.32$ and $\ensuremath{\gamma}=\ensuremath{-}{26}^{\ifmmode^\circ\else\textdegree\fi{}}$ for this nucleus. Its triaxiality is larger than that of $^{107,109}\mathrm{Tc}$, which indicates increasing triaxiality in Tc isotopes with increasing neutron number. The identification of the weakly populated $K+2$ satellite band provides strong evidence for the large triaxiality of $^{111}\mathrm{Tc}$. In $^{110}\mathrm{Tc}$, the four lowest-lying levels observed are very similar to those in $^{108}\mathrm{Tc}$. At an excitation of 478.9 keV above the lowest state observed, ten states of a $\ensuremath{\Delta}I=1$ band are observed. This band of $^{110}\mathrm{Tc}$ is very analogous to the $\ensuremath{\Delta}I=1$ bands in $^{106,108}\mathrm{Tc}$, but it has greater and reversal signature splitting at higher spins.