There is a lot of space in the edit-post-header area, particularly now with full-screen being the default setting.

I would like to add a couple of action buttons to that section. Is there already a way to do so? Perhaps a filter? I couldn't find any documentation to that effect.
For the many users in our organization, the full-screen default is going to be disorienting. I think we could mitigate that quite a bit by adding back a few of the options that are present in the regular admin header, particularly the link to create a new post.
Hey @jakeparis, there's an issue and pull request for extending the header area:
It hasn't received much action in a while. It looks neither the PR or issue discuss why the header extensibilty is needed or how it would be used. I think it's also worth remembering mobile viewports when looking at the available space, there's much less space on a phone.
Regarding the full-screen mode, there's a proposal here to make it easier to access the sidebar - https://github.com/WordPress/gutenberg/pull/21121.
But that doesn't cover the admin top bar. Would you be wiling to open up a new issue or update this one to cover that?
First off, I will say that making the WordPress logo toggle the visible state of the sidebar ( #21121 ) would be very satisfactory. I really hope that PR makes it into WP 5.5.
As far as the extensibility of the admin top bar, there is at least one _gaping_ need currently: in full screen view, there's no easy link to the public view of the page. Seeing how many of non-tech users operate, the "View Page" link is heavily used to see how their content looks in the theme context, and it's counterpart "Edit Page" then used to get back. This workflow is completely interrupted by the otherwise-very-nice fullscreen mode.
The Preview button can sort of be used in this way (and in leiu of other options probably will be), except that it's not the permalink, it's a temporary changeset url. I can be pretty sure I'm going to start seeing those urls show up in various places around our site. :(
Another way we'll use that space is to provide for a quick access to create a new post/page/cpt.
@jakeparis Thanks for the extra details. I'll rename this issue to cover accessing those items in the topbar when in full screen mode and we can treat this as an enhancement request.
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First off, I will say that making the WordPress logo toggle the visible state of the sidebar ( #21121 ) would be very satisfactory. I really hope that PR makes it into WP 5.5.
As far as the extensibility of the admin top bar, there is at least one _gaping_ need currently: in full screen view, there's no easy link to the public view of the page. Seeing how many of non-tech users operate, the "View Page" link is heavily used to see how their content looks in the theme context, and it's counterpart "Edit Page" then used to get back. This workflow is completely interrupted by the otherwise-very-nice fullscreen mode.
The Preview button can sort of be used in this way (and in leiu of other options probably will be), except that it's not the permalink, it's a temporary changeset url. I can be pretty sure I'm going to start seeing those urls show up in various places around our site. :(
Another way we'll use that space is to provide for a quick access to create a new post/page/cpt.