But what if I logged into a service on my private home network, never cleared the session and want to continue using the service from public wifi after I leave the house?
No it wouldn't make much of a difference.
What makes a difference though, is making sure the site you are visiting uses HTTPS and that your email client (if you use desktop or mobile mail client) is configured to use TLS to access your mail server.
Learn how to verify SSL certificate (no need to go for the maths, just verification based on what the browser tells you when you click the lock icon the address bar). Always verify the server's certificate, and that your are in the correct domain name before your enter your credentials.
Also, this should be obvious, but don't ignore any security warnings that the browser or the OS is telling you. If the browser says it can't make a secure TLS connection, leave your business until later.
Don't use services that handles personal information that doesn't support HTTPS/TLS while using an untrusted network, heck don't even use them at all.
You also want to make sure you don't reuse password between different websites. Especially dangerous is reusing password between sites that uses https and ones that doesn't.
You also want to make sure that your system is up to date before leaving home. If you want to update or install software, make sure you download the software over secure connection. There are ways to verify authenticity of software/files downloaded over insecure connection, by checking their cryptographic signature, but if you have to ask this question, then I'll just recommend that you just don't even try to because it's easy to get this wrong and compromise your security.
You should make sure to type https:// when accessing any sensitive websites, to prevent against sslstrip type of attack. Alternatively, bookmark the site or use HTTPSEverywhere.
All those above are difficult to do consistently, and it easy to make mistakes. For added security, you may want to enable TOTP Two Factor Authentication (e.g. Google Authenticator app) for services that provides them, so that even if you slipped and someone swiped your password, they'll still be unable to access your account without also stealing your second factor, usually your mobile phone. Many major web services now supports 2FA, e.g Dropbox, Google.