Children under age 16 (until one day before they turn 16, e.g. 15.99 years) always travel with 50% discount. You do not need to add your daughter to the Swiss Half Fare Card
Children under age 16 need a ticket for CITY A to CITY B like Half Fare Card, even when they do now need the Card, they just are treated as reduced.
Here is a local’s pro hint: If you just stay for some days, the Day Pass Children aged 6-16 may fit your needs. For just 19 Swiss Francs, your dauther can use it to travel through Switzerland alone, together with other children or with adults for a whole day. The Day Pass Children can be purchased online, see the above link.
If you stay a few weeks, maybe the Junior Travelcard might be an option for you. With the Junior Travelcard, children aged 6 or over and under 16 can travel throughout Switzerland free of charge for a whole year when accompanied by a parent. The travelcard costs only 30 Swiss francs for a year, must be issued onsite at a ticket counter and is delivered on the red SwissPass Card.
I wish you a wonderful stay in Switzerland.