So there's a slight cognitive dissonance here. Outside of an architect role - the only real way up is taking on more management tasks.
A lot of larger companies are seeing this as an issue so they are adding gradients to what a SSE is (SSE 1, SSE 2 etc) but really the answer is you're going to have to move up via a team leadership role and this means more management and less engineering.
A think it depends on what you mean by "management task." If you take the initiative to drive a really valuable project, can assume a leadership role while ignoring a lot of the tasks the company typically assigns to a manager, like performance reviews and meetings and ticket/bug metrics and timesheets and project plans and timelines and deadlines and fielding every request that comes your group's way.
Meanwhile a good leader will drive and inspire the team by making and pushing excellent design decisions, obtaining needed resources, doing the oxen's share of grunt work to get the project off the ground, and energetic mentoring of other developers.
In the places I've worked that were large enough to actually have names and levels and whatnot, manager/non-manager was always a totally separate dimension from job level. You could easily have a level N engineer being managed by a level N-1 manager. A level N engineer becoming a manager produced a level N manager, it was not a promotion.
So I've never seen how becoming a manager means going "up."
A lot of larger companies are seeing this as an issue so they are adding gradients to what a SSE is (SSE 1, SSE 2 etc) but really the answer is you're going to have to move up via a team leadership role and this means more management and less engineering.