Best practice for overflow menus

We have a bottom bar menu in one of our mobile applications with an overflow icon which currently opens the overflow menu.

Here is what it currently looks like:

We are updating our application UI and have proposed a new look and feel and mechanism for the menu. The new proposal is as follows: when the user selects the overflow icon on the far right of the bottom bar, the menu will slide in from the left and display over 80% of the page. My concern is that it might be a bit odd when the user selects the overflow icon on the right and the menu slides in from the left.

New layout:

My question is - should the overflow menu rather be a full page that appears and does not slide in from any side, or should it remain an overlay but perhaps slide from the bottom bar and not the left of the screen?