There are many ways to get alerts to new issues as they are published. The difficulty is in connecting to full text where subscriber access is required.
Note that some journals embargo their current content (anywhere from 6 months to a couple of years). In that time it is available only to subscribers but becomes freely available the end of the embargo period. If a journal provides a relatively poor benefit (in terms of the cost per article over an embargo period), the Library may choose to acquire articles in other ways for that period of time.
Use the BrowZine app - the app will generate notifications as new issues are published.
Set up an Ovid Discovery account, log in, search for a journal title in the Library eResources tab, Activate Journal Alert. The emailed alerts that will come through will incorporate the authentication to get you past paywalls.
Set up alerts yourself on publisher sites
To connect to subscriber-restricted full text though, you need to use one of the Library's nifty tools to convert the links you land on to ones that will let you through. Try a bookmarklet or the LibKey browser extension.
What if the Library doesn't subscribe to a title you get a table of contents for?