Apologies if you are hardcore VS Code with no intention to switch.
I find IntelliJ's handling of type inference to be amazing. It displays inferred types (local variables, args to | ... | { ... }'s) in blue as some type of "label" (so you can see it, but the cursor skips over it when navigating).
I don’t know if the official rust extension can offer what you want since I never really used it myself. You might like to try out Rust analyzer as an alternative; works great with vscode and does definitely offer type information when hovering over variables (or any kind of expression for that matter).
I found rust-analyzer very helpful comparing to default extension but it was not catching most of my projects in sub-directories
I added entry to settings.json