I think that after this saga Blackbeard next move will be attack Big Mom and get her territory.
If we think about Blackbeard during the story, he was always very patiente, he is not against taking risks, but he rather do it so when they are the lowest possible, in other words he always waits the perfect moment to make his moviments.I also think that is clearly that he wants to be the next Pirate King, not being content with his current position and to do that he needs to defeat Big Mom and get her Poneglyph.
To attack someone as powerfull as Big Mom I think Teach would wait until a moment that he is at his best and she is at her worst, and I think after this saga will be this moment.
He is at his best because of two things: first, he has just acquired a lot of weapons from his attack against the Revolutionary Army; second, most likely Aokiji has just joined his crew, in the last 2 years Blackbeard was probably never this powerful.
Big Mom will be at her worst at the end of this saga for various reasons.
Second, her crew power will very diminish at the end of this saga. She lost a lot of foot soldiers, a lot of her ships were damaged, she lost her protection net, and at least one of her commander is down (and as we saw in the war, commanders play a big role in those full scale war) and Perospero (who is probably her best fighter outside the commanders) is very heavily injured. And that’s the best case scenario considering today, if Luffy defeats Katakuri and Sanji, Bege or the Vinsmokes take out other big players her crew will be probably at the weakest point it ever was.
Think about it, before this saga Teach could probably tie in a fight with Big Mom, and his best fighters also tie with the best of Big Mom, but, now imagine if he and Aokiji go head to head against a weaken Big Mom. And if the Big Mom Pirates have to defend themselves against Blackbeard’s Commanders without lets say Oven, Perospero, Cracker AND Katakuri? It would be without a doubt the best moment for Blackbeard to attack.
Also I think after this saga Oda will be done with the Big Mom Pirates (the pirate crew I mean, maybe some characters will survive and make an appearance in the future, maybe even Big Mom herself). Oda rarely (or maybe even never) visit the same place twice with the Straw Hats, so I don’t think they will ever return to Totto Land. And I also don’t think that we will see a full scale war, which is reserved for Kaido, against Big Mom because it would be kind repetitive. Furthermore, Teach defeating Big Mom would elevate even more his position as Luffy’s main antagonist.
So those are my reasons to think that after this saga, Blackbeard will attack and defeat the Big Mom Pirates.
*Theory by leanderbanegas