However I do think you're correct in saying that Fort Randolph and Fort Wright (at Randolph, Tenn.) appear to have been the same fort. This is from a thesis by Thomas Lee Anderson in which he talks about W.T. Sherman's orders to burn what was left of the ghost town of Randolph in response to a Confederate guerrilla attack on a Union packet boat on the Mississippi.
Anderson says:
"Following the opening salvos on Fort Sumter, in April 1861, a letter from an unnamed Tipton County resident appeared in the Memphis Appeal suggesting that state authorities send troops and artillery to Randolph. In the letter the author referred to Randolph as a “near deserted village that was once the mighty arch-rival of Memphis.” Randolph was, the writer claimed, the perfect place, high on the Chickasaw bluffs, from which to defend Memphis from an attack by Union forces on the Mississippi. Tennessee Governor Isham Harris promptly dispatched Lt. Col. Marcus Wright of the 154th militia regiment at Memphis to Randolph where, on the site of the “near-deserted village,”
Fort Wright/Fort Randolph was constructed. Some of the town’s derelict buildings provided lumber for the construction of warehouses, while
an underground powder magazine was dug out of the banks of the Mississippi (which you provided photos of above, thanks buddy!). By early May it was reported in the Memphis Daily Appeal that 400 men were in training at Randolph. At the end of May, soon-to-be Confederate Generals John Sneed and Gideon Pillow hosted a visit to Fort Wright from several Memphis-area ladies who were “sumptuously entertained.” During the summer of 1861 officials from the Confederate national government came to Tennessee to take control of the troops and the defenses. There was a great hue and cry from concerned citizens worried about the loss of local control, but they were mollified by both Confederate and Tennessee officials who assured them of convergent interests.
In July Fort Wright was closed; the troops and equipment were
moved upriver to Fort Pillow. Randolph became once again a near-deserted village along the Mississippi.
Sherman, says Walters, “exploded into action” when he heard the report of the attack on the
Eugene. By nightfall on September 24, 1862, the
Ohio Belle and the
Eugene were filled with the Ohio 46th Volunteer Infantry along with a battalion of artillery. Sherman had suggested to Col. Walcutt, whom he placed in command of the expedition, that he send one boat past Randolph to see it would draw fire; if it did, Walcutt and his troops would know then what they were up against at Randolph. The flotilla reached the area before daybreak on September 25. The
Ohio Belle landed Walcutt and his troops below Randolph while the
Eugene steamed up the Mississippi as far as Fort Pillow without drawing any fire. Meanwhile Walcutt and his troops reached Randolph without resistance. They found no town, only a mostly deserted village with six houses and
dozens of abandoned and derelict buildings left over from Fort Wright and from older projects at Randolph."
http://digitalcommons.wku.edu/cgi/viewcontent.cgi?article=1088&context=theses
Which means there was no "Fort Randolph" there, that Fort Wright had been the only fort there and when it was abandoned there were
no defenses remaining in the Randolph area. Certainly had there also been a Fort Randolph in addition to a Fort Wright then wouldn't at least some of Fort Wright's garrison have been moved there instead of Fort Pillow when Wright was abandoned? And Anderson makes no mention of the Union troops finding evidence of a second fort there, just a near-deserted village and the remains of the abandoned Fort Wright. So it sounds pretty certain that these "two" forts were really the same fort as you said.