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Excited states with seniority $\ensuremath{\nu}=3$, 5, and 7 have been investigated in odd neutron-rich $^{119,121,123,125}\text{Sn}$ isotopes produced by fusion-fission of 6.9-MeV/$A\phantom{\rule{4pt}{0ex}}^{48}\text{Ca}$ beams with $^{208}\text{Pb}$ and $^{238}\text{U}$ targets and by fission of a $^{238}\text{U}$ target bombarded with 6.7-MeV/$A\phantom{\rule{4pt}{0ex}}^{64}\text{Ni}$ beams. Level schemes have been established up to high spin and excitation energies in excess of 6 MeV, based on multifold gamma-ray coincidence relationships measured with the Gammasphere array. In the analysis, the presence of isomers was exploited to identify gamma rays and propose transition placements using prompt and delayed coincidence techniques. Gamma decays of the known $27/{2}^{\ensuremath{-}}$ isomers were expanded by identifying new deexcitation paths feeding $23/{2}^{+}$ long-lived states and $21/{2}^{+}$ levels. Competing branches in the decay of $23/{2}^{\ensuremath{-}}$ states toward two $19/{2}^{\ensuremath{-}}$ levels were delineated as well. In $^{119}\text{Sn}$, a new $23/{2}^{+}$ isomer was identified, while a similar $23/{2}^{+}$ long-lived state, proposed earlier in $^{121}\text{Sn}$, has now been confirmed. In both cases, isomeric half-lives were determined with good precision. In the range of $\ensuremath{\nu}=3$ excitations, the observed transitions linking the various states enabled one to propose with confidence spin-parity assignments for all the observed states. Above the $27/{2}^{\ensuremath{-}}$ isomers, an elaborate structure of negative-parity levels was established reaching the ($39/{2}^{\ensuremath{-}}$), $\ensuremath{\nu}=7$ states, with tentative spin-parity assignments based on the observed deexcitation paths as well as on general yrast population arguments. In all the isotopes under investigation, strongly populated sequences of positive-parity ($35/{2}^{+}$), ($31/{2}^{+}$), and ($27/{2}^{+}$) states were established, feeding the $23/{2}^{+}$ isomers via cascades of three transitions. In the $^{121,123}\text{Sn}$ isotopes, these sequences also enabled the delineation of higher-lying levels, up to ($43/{2}^{+}$) states. In $^{123}\text{Sn}$, a short half-life was determined for the ($35/{2}^{+}$) state. Shell-model calculations were carried out for all the odd Sn isotopes, from $^{129}\text{Sn}$ down to $^{119}\text{Sn}$, and the results were found to reproduce the experimental level energies rather well. Nevertheless, some systematic deviations between calculated and experimental energies, especially for positive-parity states, point to the need to improve some of the two-body interactions used in calculations. The computed wave-function amplitudes provide for a fairly transparent interpretation of the observed level structures. The systematics of level energies over the broad $A$ = 117--129 range of Sn isotopes displays a smooth decrease with mass $A$, and the observed regularity confirms most of the proposed spin-parity assignments. The systematics of the $B(E2)$ reduced transition probabilities extracted for the $23/{2}^{+}$ and $19/{2}^{+}$ isomers is discussed with an emphasis on the close similarity of the observed $A$ dependence with that of the $E2$ transition rates established for other $\ensuremath{\nu}=2$, 3, and 4 isomers in the Sn isotopic chain.