Oh I hear you, it’s always an intriguing question: what else could have been done? Certainly Winfield Scott thought “more scalpel than sledge hammer” was the trick - that was the basis of his original Anaconda Plan. But even in order to “just” seal off the Confederacy by land as well as sea, a large army would still be required to watch the long border. Where would it come from?
You wouldn't need very many. They had roughly 16,000 and you could have plenty dispersing them to key points. Maybe 2,000 in Memphis, (need artillery), 2,000 in Chattanooga (just fortify the rail depot and plant a battery backed up by a couple companies on a mountain), maybe 1,200 in North Carolina at Charlotte and Wilmington. Do that the rail lines are completely cut off, and the Mississippi at Memphis with enough artillery.
Excepting Arkansas, in places like Tennessee there was something of a disdain of Southerners from the Deep South. I would think that would have opened an opportunity to purposely, (but very secretly) turn a blind eye to Confederates from Alabama and Mississippi raiding into Tennessee to hit Union targets and violate the State, to tick off Tennessee into calling up their militia to combat them, thus more troops. If the theoretical tricking the CSA into violating those States borders worked, after two or three months Lincoln probably could've got away with calling up 75,000 volunteers politically in the Upper South.
Heck given its proximity to Tennessee, and how it was incredibly dependent on rail trade north of it, after a couple weeks US troops probably could've occupied Corinth before CS militias and volunteers could be made into an effective fighting force, (a lot harder without those 4 Upper South States), claiming the people wanted them to in the newspapers, and forcing your 1st Manassas there from behind fortifications with nothing but Regulars and ticked off Tennessee militia.
The entire key would be to keep the CSA looking like the aggressor at all times. But at the end of the day its a "what if" after all.