I have recently added an SSD to an old Raspi 3B which is running MariaDB and my live HA on a Raspi 3B+ connects to it over my home network.
Here is a guide that I wrote at the time of the install that may be useful to follow
1, Download Raspbian Stretch Lite from Raspberry Pi and flash to the SSD using Etcher
https://downloads.raspberrypi.org/raspbian_lite_latest
Connect the SSD via USB to any USB port on the Pi, connect Pi up to TV and turn on the power. Raspian should boot shortly and the Pi will resize the SSD. To connect to your network use an ethernet cable or connect to WiFi on the top right of Raspbian entering your home SSID/PW.
2, You will need to enable SSH on the Pi as well, guide here :-
Once SSH is enabled connect to the Pi using Putty username & password are pi and raspberry - good time to change the default password using “sudo passwd”
3, Install Docker - The easy way to set up Docker on a Raspberry Pi
curl -fsSL get.docker.com -o get-docker.sh && sh get-docker.sh
4, Install HA - GitHub - home-assistant/supervised-installer: Installer for a generic Linux system
sudo apt-get install jq // you may need to install jq first
sudo mkdir /usr/share/hassio // create container
sudo su // enable superuser
curl -sL https://raw.githubusercontent.com/home-assistant/hassio-installer/master/hassio_install.sh | bash -s – -m raspberrypi3
After a while Hassio should start the full install and be available in a browser via its IP address on port 8123.