Often, advancement into better school bands is available if a student is willing to change to tenor or baritone sax. Many second instruments are purchased to facilitate this need, and the process usually grows the young musician as they learn another, very similar instrument. It adds to the list of things they can do and increases their value to the musical organization in which they participate.
Another opportunity that can imply branching out into other saxophones or even other instruments in the woodwind family is jazz band. The jazz band is often one of the coolest groups to play for in school. With the more difficult music and a more exposed setting, many saxophone players find a need to not only upgrade their existing instrument, but, often find a seat open for a saxophone they don't play yet. They might also find some flute or clarinet parts in that jazz band music book.
The best way to prepare is to actually prepare! Opportunity is great when you are ready for it. What I suggest is, start your saxophone player on a double early. If you find you student is doing really well on sax and performing at a solid intermediate level, maybe even complaining that the music is too easy at school, hand them a clarinet or flute.
I started playing clarinet when I moved onto high school. My freshman year, I was handed a clarinet and told to learn how to play it. I practiced a bunch and found my sophomore year started with a flute assignment. The fact of the matter is, jazz band music has parts in the music for clarinet and flute. Both of the instruments I started on were school instruments, but it was not long before I upgraded both with purchases. I was much happier with a new student flute and found a great deal on a wood clarinet.
I was really lucky because the school instruments in junior college played really well. I didn't have to think about purchasing a baritone saxophone or bass clarinet until later in life when need dictated action. We didn't have a whole lot of money when I was in school. We just saved and bought, as we needed to, over the years.
Another awesome sax to consider is the soprano saxophone. Although there are not as many opportunities, the soprano is a really fun (clarinet looking) sax to add to the closet. The only person that 'needs' a soprano sax in jazz band is the lead alto player. The alto chair can have soprano sax parts in many jazz band charts.
For me, I played mostly tenor and my interest didn't develop for soprano sax until I was in college and found there were musical jobs I would have to pass on if I didn't get a soprano sax. I am so glad I purchased a great soprano sax. I perform on my soprano every day, and people love the sound of that sax.
You may find that much of the inspiration for a new instrument will come from listening to the sound of a sax and falling in love with the possibilities.The best place to start is to talk to your band director and be aware of the opportunities that might lie ahead. A new instrument might inspire a whole new world of musical possibilities and open some doors along the way.