The army with which Thomas fought the Battle of Nashville two weeks after Franklin was about 50,000 strong. Of these, about 20,000 were Schofield's men who had fought at Franklin and two divisions (IIRC) of about 10,000 men were from Missouri and did not arrive at Nashville until after Franklin. The remaining 20,000 were generally inexperienced and not well trained and the cavalry lacked horses.
Good enough to hold fortifications, though.
So, assuming that Hood somehow pulls off a magical victory at Spring Hill and captures Schofield's entire force without much loss to himself, his 35,000 hardened veterans would be faced with 20,000 less experienced men at Nashville. But the defenses of Nashville were probably stronger than any other city in Federal hands aside from Washington itself and reinforcements were on their way. Even if he had a best-case-scenario that allowed him to deploy superior numbers and better troops against Thomas, I don't see Hood gaining control of Nashville. Its fortifications were just too strong.
And Thomas too good a commander - even in the miracle scenario, the AoT is going to be facing at best 3 to 2ish odds in its favor, which would take either troops or commanders or both that would be easily stampeded to be confident of beating even if we're talking no fortifications at all.
I can only assume Hood operated on the principle that he would do all he could, and God would supply the rest - because otherwise him being high as a kite at Franklin and/or raging mad are less insulting to his character than the idea that he was lucid and tried the 1% chance of success tactic (possibly the best if Franklin had to be fought, but it only had to be fought because Hood launched this loony campaign in the first place) anyway.
And let's say - somehow - Hood manages to bag Nashville's garrison, with his army still at 30,000 strong.
Somehow. Angels come down and fight for the Confederacy. A handful of a-bombs land on Nashville. Thomas gets replaced by a resurrected Dixon Miles. Whatever it is, Hood gets his miracle.
That's almost entirely meaningless in accomplishing anything with any impact on the war. Hood is far out in left field so far as even temporally interfering with any major Federal operations, and that's the good news for the poor AoT.