After posting the comment above, my SE got back to me with another solution to this problem that seems to work better than the one I posted above. Its written by Graham at F5 who specializes in SAML. https://devcentral.f5.com/s/articles/apm-cookbook-autolaunch-saml-resources-21377
 
Here are Graham's comments to our SE regarding the solutions by Kunjun and Milkman (Milkman posted the solution I referenced in the other thread I linked to in my previous comment) and why his solution is more complete:
 
"[Their solution] only handles the access policy completed event so if they later come back to the existing session it will not fire because that event isn’t hit, that’s why mine has two events to cover the two access scenarios. Also [they assume] you always want the user redirected to the same SAML resource, what if you have multiple, that’s why mine leverages a switch."
 
The benefit of Graham's solution for me was that with Milkman's I had to lower the timeout threshold, because if you closed the site you accessed through SAML, you couldn't access it again unless the previous session was ended. With Graham's solution you can access it even if the previous session isn't closed. He described that scenario above.