Local hostnames for Docker containers

Another solution would be to use a browser with a proxy extension sending the requests through a proxy container that will know where to resolve the domains to. If you consider using jwilder/nginx-proxy for production mode, then your issue can be easily solved with mitm-nginx-proxy-companion.

Here is an example based on your original stack:

version: '3.3'

services:

  server:
    build: ./server
    working_dir: /app
    volumes:
      - ./server:/app

  client:
    environment:
      - VIRTUAL_HOST: client.dev
    image: php:5.6-apache
    volumes:
    - ./client:/var/www/html

  database:
    image: postgres:9.4
    restart: always
    environment:
      - POSTGRES_USER=postgres
      - POSTGRES_PASSWORD=root
      - POSTGRES_DB=dbdev
      - PG_TRUST_LOCALNET=true
    volumes:
      - ./database/scripts:/docker-entrypoint-initdb.d # init scripts

  nginx-proxy:
    image: jwilder/nginx-proxy
    labels:
      - "mitmproxy.proxyVirtualHosts=true"
    volumes:
      - /var/run/docker.sock:/tmp/docker.sock:ro

  nginx-proxy-mitm:
    dns:
      - 127.0.0.1
    image: artemkloko/mitm-nginx-proxy-companion
    ports:
      - "8080:8080"
    volumes:
      - /var/run/docker.sock:/var/run/docker.sock:ro

• Run docker-compose up
• Add a proxy extension to your browser, with proxy address being 127.0.0.1:8080
• Access http://client.dev

The request will follow the route:

• Access a local development domain in a browser
• The proxy extension forwards that request to mitm-nginx-proxy-companion instead of the “real” internet
• mitm-nginx-proxy-companion tries to resolve the domain name through the dns server in the same container
  • If the domain is not a “local” one, it will forward the request to the “real” internet
  • But if the domain is a “local” one, it will forward the request to the nginx-proxy
• The nginx-proxy in its turn forwards the request to the appropriate container that includes the service we want to access

Side notes:

• links removed as it's outdated and is replaced by Docker networks
• you don't need to add domain names to server and database containers. client will be able to access them on server and database domains because they are all in the same network (similar to what link was doing previously)
• you don't need to use ports on server and database containers because it only forwards ports to be used through 127.0.0.1. PHP in client container will do only "back-end" requests to other containers, and because those containers are in the same network, you already can access them with database:5432 and server:3000. The same goes for server <-> database connections.
• I am the author of mitm-nginx-proxy-companion

My Bash script WITH ALIAS without docker-machine Based on http://cavaliercoder.com/blog/update-etc-hosts-for-docker-machine.html

#!/bin/bash

#alias
declare -A aliasArr
aliasArr[docker_name]="alias1,alias2"

# clear existing *.docker.local entries from /etc/hosts
sudo sed -i '/\.docker\.local$/d' /etc/hosts

# iterate over each machine
docker ps -a --format "{{.Names}}" \
| while read -r MACHINE; do

    MACHINE_IP="$(docker inspect --format='{{range .NetworkSettings.Networks}}{{.IPAddress}}{{end}}' ${MACHINE} 2>/dev/null)"
    
    if [[ ${aliasArr[$MACHINE]} ]]
    then
        DOMAIN_NAME=$(echo ${aliasArr[$MACHINE]} | tr "," "\n")
    else
        DOMAIN_NAME=( ${MACHINE} )
    fi

    for addr in $DOMAIN_NAME
    do
        echo "add ${MACHINE_IP} ${addr}.docker.local"
        [[ -n $MACHINE_IP ]] && sudo /bin/bash -c "echo \"${MACHINE_IP} ${addr}.docker.local\" >> /etc/hosts"
        export no_proxy=$no_proxy,$MACHINE_IP
    done
        
done