Actix_web MongoDB docker-compose CI/CD and dev workflow

I'm buidling a personal project of web application for which I need to create a little REST API service. I chose to use Rust but it's my first experience with Rust on a web server, I'm more used to things like Python PHP, etc.
I also chose to containerize my API with docker and I can't figure out how am I suppose to do to have a fluid development workflow and a efficient deployment process.
Most of the tutorials I read says that it's a good practice to use a multistage building of the docker image in order to compile the code in an intermediate docker layer and then create the final image only with the compile binary. But I don't understand how am I supposed to do to update the code without having to recreate the entire image or at least stopping the container.

So for now I tried a temporary solution: I have a full rust:buster-slim image, my source code is bound into the container with a volume and I compile and execute the binary into the container wih cargo run the same way I would do it ouside the container. It takes way less time than rebuilding the image because I don't lose the compilation cache and the container remains the same.
This solution is working for now because I'm only developing locally. But I don't think I could use that in production because I have to stop and restart the container for each upgrade of my api.

So my request for help is about which concepts and type of architecture should I look after in order to have a dynamic development and deployment workflow.

This is my current docker-compose and Dockerfile:

# docker-compose.yml
version: "3.9"
        build: .
            - mentalodb
        container_name: "rust-api-container"
        restart: unless-stopped
            - "8080:8080"
            - ./src:/usr/src/rust_api/src
        command: cargo run --release

        image: "mongo"
        container_name: "mongo-test-container"
            - MONGO_INITDB_DATABASE=mentalodb
            - MONGO_INITDB_ROOT_USERNAME=********
            - MONGO_INITDB_ROOT_PASSWORD=*********
            - ./init-mongo.js:/docker-entrypoint-initdb.d/init-mongo.js:ro
            - /var/mongo-volume:/data/db
            - "27017-27019:27017-27019"
# Dockerfile
FROM rust:1.50-slim-buster
WORKDIR /usr/src/rust_api
COPY ./Cargo.toml ./Cargo.toml
COPY ./src ./src

And I reload the api if I make changes in the code simply by doing docker-compose stop rust_api && docker-compose up rust_api, wich triggers the cargo run command.
So this is a simplistic temporary solution but I don't understand how I should do it the right way.