I propose to move the search bar up into into the new nav bar. Similar to this:

• It's more reasonably placed closer to the other navigation elements. The placement in the side bar historically comes from the sphinx theme.
• This also declutters the sidebar.

Attn. @dorafc, you might want to consider this for the new front page layout. I could hack this together, but for sure you know better how to lay this out properly.