Working with other middle managers to align resources for upcoming projects according to company goals. For example, if you know you need to implement a foobar next quarter that will require another team to implement a bazblee that you depend on, they can work with the bazblee senior manager to make sure the bazblee gets done, instead of the plumbus being requested by another group (that is, if the foobar and bazblee are more important to the company's goals than the plumbus - otherwise, they'll do the plumbus and you won't get your bazblee - so you probably shouldn't work on your foobar project)

Convincing other managers of your project's alignment with company goals is usually not an exact science, so your middle manager's ability to get these things committed to and done can be highly variable, depending on their connections with other middle managers and their ability to influence.

You can tell if you have a good middle manager if things you need from the rest of the company tend to fall into place, versus constantly being disappointed when you ask for stuff from other groups in the company.

Of course this all requires skillful ICs and team managers to manage up and make sure the middle managers know what they are doing and what they will need from the rest of the company.