How to carry a trumpet for long trips
I'd suggest a top-of-rack bag for your instrument, because its more aero when the load is in-line with the rider.  If there's no bag big enough, use a series of bungee cords to securely strap the case to the top of the rack, lengthwise.  Feed a cord under the case's handle too.   If this becomes a regular thing, you can even make a &quot;tray&quot; permanently secured to the top of the rack, exactly the right size for your case, with lips/sides to stop it sliding out.
You could use panniers / saddle bag on the side of your rack, but that tends to stick out into the wind a big more.  There's also the increased risk of damage being stuck out.  But a pair of panniers lets you carry other stuff on the other side, and still use a top rack bag/tray.
Carrying the trumpet on your back is a workable solution too, but it increases the sweatiness of your back, and can exacerbate damage and injury in the event of an accident so its not ideal.
Carrying it inside the frame triangle, or above the top tube is probably not going to fit, and will interfere with your legs while pedalling.
A front rack might be a good idea too - you can see it while riding.
A trumpet is not really large enough to need a trailer, so that can wait until you upsize :-)

Whatever you choose, also consider waterproofness.  Most instrument cases have a velvet / velveteen liner fabric which doesn't like getting wet.  If you have to ride in the rain, consider a &quot;dry sack&quot; large enough to hold the whole case and seal up.   Good for a towel and dry clothes too.  This would also help prevent marking of your case by vibrations against the rack.