ReAuth for Forge supports Minecraft 1.7.10–1.19.3 and lets you log back in after an invalid session without restarting the game. Check your session status or switch accounts while Minecraft stays open. This is a client-only mod; placing it in a server’s mods folder has no effect.
For your rented Minecraft server, one-click installation is available from the panel, but ReAuth must run on your client to help you reconnect.