Each left the field separately about midday via the Dry Valley Road and each eventually ended up in Chattanooga. While retreating as far as Rossville was inevitable, their decisions to ride to Chattanooga, made individually and for a variety of reasons, damaged each man's career.
Rosecrans' encounter with Chief of Staff James Garfield famously resulted in Rosecrans hurrying on to Chattanooga while Garfield returned to the battlefield to find the embattled Maj. Gen. George Thomas. McCook forced a civilian guide to lead him, and ended up bypassing Rossville entirely. Crittenden tried for a time to rally stragglers along the Dry Valley Road before telling a member of his staff, "I believe I have done all I can." [Powell and Friedrichs, page 250]
The Ninth and Seventeenth Kentucky Infantry made decisions at the regimental level, joined by other officers and men of their own brigade and were indeed detached from their brigade commander who was fleeing with (or trying to rally) most of his men. Following their exact movements after dark is difficult. It is certain that they were not with Brig. Gen. Sam Beatty, but most likely remained with Brannan's Division of Thomas' XIV Corps. This assumption is based solely on their proximity and later correspondence, since Gen. Brannan failed to mention them is his official report. Powell and Friedrichs state that "One division, Van Cleve's of the XXI Corps was so badly disrupted that it could not be reformed even in skeleton form until the next morning." [page 250] Whether this skeleton included Sam Beatty and his stragglers and or the Ninth and Seventeenth (et al) survivors from Snodgrass Hill is unclear, but it is likely that Col. Stout and his men remained with Brannan's division on the night of the 20th as defensive positions were being created.
Note: The following excerpt from Col. Stout's report dated Sept. 24, 1863 provides clarification:
When dawn on the 21st arrived, the Army of the Cumberland was in surprisingly good shape considering the disaster that had befallen it the day before. Most of it's formations were intact, if still suffering from straggling, and Thomas had been able to refill his cartridge boxes from ammunition found or moved to Rossville. The Army also had an imposing defensive position in Missionary Ridge. Bragg would be hard-pressed to simply smash through Rossville Gap by sheer force. Any formal attack there would likely cost the Confederates a great deal more men. [Powell and Friedrichs, page 250]