Sadly what you are experiencing is how snapping works currently ... the snapping environment is only changed by a selection made on the Targets tab. Of course you can open the Esri snapping window and override manually. The best workaround is to right-click the feature on the selection tab and choose "Send to Targets" this will set the snapping for you.
There is an open enhancement request (idea) to support the Selection tab. Vote it up if you think it is worthwhile Snapping Rules Need Improvement
Thanks Neil. Not really what I wanted to here.