The ice-albedo feedback on rapidly-rotating terrestrial planets in the habitable zone can lead to abrupt transitions (bifurcations) between a warm and a snowball (ice-covered) state, bistability between these states, and hysteresis in planetary climate. This is important for planetary habitability because snowball events may trigger rises in the complexity of life, but could also endanger complex life that already exists. Recent work has shown that planets tidally locked in synchronous rotation states will transition smoothly into the snowball state rather than experiencing bifurcations. Here we investigate the structure of snowball bifurcations on planets that are tidally influenced, but not synchronously rotating, so that they experience long solar days. We use PlaSIM, an intermediate-complexity global climate model, with a thermodynamic mixed layer ocean and the Sun's spectrum. We find that the amount of hysteresis (range in stellar flux for which there is bistability in climate) is significantly reduced for solar days with lengths of tens of Earth days, and disappears for solar days of hundreds of Earth days. These results suggest that tidally influenced planets orbiting M and K-stars that are not synchronously rotating could have much less hysteresis associated with the snowball bifurcations than they would if they were rapidly rotating. This implies that the amount of time it takes them to escape a snowball state via CO\(_2\) outgassing would be greatly reduced, as would the period of cycling between the warm and snowball state if they have a low CO\(_2\) outgassing rate.