If you only get a discount, then you don't need to use a travel day.

NOte-this ONLY applies to the list of other railways in Swiss that are listed in the official info. Most USAers seem to want to visit Interlaken-Grindelwald and around-these, quite expensive, railways do not accept passes for free, only for discounts (and then making it expensive still).

When people go searching in various Swiss timetables etc. they will find prices for ´half fare card holders´-about half of all Swiss indeed do have such a card or the like. Your EUrail will NOT be accepted as such! F.e. there are also daytickets for all of Swiss-if ordered advance with quite a big discount and valid on all transit-also bus, tram etc

Without more details this remains-as so often-guesswork.

Also note that in many places in Swiss your HTL-and even hostel-may give free passes for local transit (mostly only tram-bus)

Which lines are you thinking of? If staying in a particular area for a few days a local full price pass may be cheaper. As above you don’t use a travel day for discounted travel