You are raising a frame challenge, I see. The question asks for tactical role and you're rebelling against it and proposing a strategical/logistics one.

You can not assume they have sufficient history knowledge of the war to provide advice.

I question how a group of time travelers gets an army to listen to them long enough to lecture them on topics like strategy and squad tactics. Militaries tend not to be easily swayed by new ideas even when they come from within, let alone from strangers. (The logistics and medical corps may be easier to persuade - they're used to inventors and industrialists coming to them with ideas, although they're also used to those ideas not panning out.)

@Cadence I think that when they show off their automatic weapons and radios, people are going to pay attention rather quickly.

Speaking of knowledge of the war... How about detaining any general that defects. Lee was tempted to fight for the Union, but said he couldn't fight against his home state. If Lincoln had just detained him, the war would have gone a lot better for the Union.

Do you want confederates on the moon? Because this is how you get confederates on the moon!

@Mary Knowledge of the specific war doesn't matter. Knowledge of modern tactics and warfare techniques would be plenty valid and you can safely assume highly trained soldier have been highly trained in being soldiers.

Although true machine guns would be a huge shock, do note that the first gatling gun saw (limited) deployment during the Civil War, and canister shot had been invented during the Napoleonic wars.

@StevenGann The problem is that modern tactics only work because of modern technology. Muskets do not work with modern tactics. Period. You might be able to improve things by using some older stuff like Lanchester’s laws to great effect, but modern tactics are at best impractical when your accuracy and fire rate are bad enough to necessitate volley fire tactics.

@AustinHemmelgarn The problem is the soldiers aren't armed with muskets they're armed with modern weapons. You might consider reading the post.

@StevenGann They're referring to Union soldiers being taught modern tactics.

@StevenGann The answer offered their specific knowledge of the war as a selling point. But they would need it to sell it.

Accurately predicting the future might not be a good idea. They would be accused of being witches and burned on the spot.

@laolux That was the 1600s, not the 1800s.

@KEY_ABRADE I know, most of them were before the US American civil war, but at least in Salem they had another witchcraft trial in 1878. I can well imagine that the outcome of that trial would have been different if there was better evidence such as magic long distance communication devices.

They appear out of nowhere and bring perfect intelligence? Sounds like spies to me. I wouldn't trust them and the better their predictions of enemy movements, the more I would suspect eventual betrayal.

@Falco Spies don't come with super-fast-firing guns and magical communications devices. Nor do they have fifty stars on their version of the American flag. Nor do they include black people. Nor can they predict historical events unrelated to the war.

@StevenGann Given the comment you replied to, my assumption was that your comment was about the soldiers ability to act as tactical advisors, hence my comment about the applicability of modern tactics to civil-war era armies (I also now realize my comment was a bit anachronistic, muskets were kind of old tech by the Civil War, but the rifles they did have then still wouldn’t work very well with modern tactics).

"Training them in modern hand-to-hand combat would also be quite useful." - this is false. Most modern soldiers have almost no hand-to-hand training. If anything, the historical soldiers are probably more used to knife / close combat.