There are a whole slew of trumpets pitched between the standard B-flat or C Trumpets and the Piccolo Trumpet. These are the Sopranino Trumpets. There are instruments pitched in G, F, E, E-flat, and D, all sounding above written pitch. Like the interchangeable slides for the Piccolo Trumpets, several of these instruments are really just a single instrument like an F/G Trumpet or an E-flat/D Trumpet (the E Trumpet may be a third set of slides for either instrument, usually the F/G instrument). Of these two instruments, the F/G Trumpet is far rarer, but most professional trumpet players will possess an E-flat/D. The rare E Trumpet is generally only used for the Hummel Concerto, which was originally composed in E, but is usually performed in E-flat. Choice of which instrument to use is entirely up to the player. Even with the best of intentions, a composer’s wish will usually go unheeded. The player will simply choose the instrument which will make the passage easiest and give the best effect.
In general, the higher pitched the instrument is, the thinner and smaller the sound will be. So the High F Trumpet will be smaller and thinner in sound than the D Trumpet.
In my own writing, I generally would only use the E-flat Trumpet. It is far enough away from the B-flat and C Trumpet to have an audible difference in timbres, and it is common enough for most players to carry one with them.
In sound, these trumpets are closer to the B-flat and C Trumpets than they are to the Piccolo Trumpet. They have a very clear strident sound, but not as clear as the Piccolo Trumpet. For a good example of how one of these trumpets sound, listen to a modern recording of the Haydn Trumpet Concerto which is written an E-flat Trumpet (though the original was written for keyed trumpet).
In all other respects, the Sopranino Trumpets are identical to the C and B-flat Trumpets. They usually possess three valves (though some E-flat Trumpets will have four), so they will only descend to the written F-sharp. Remember what I said about the Piccolo Trumpet and high notes. These higher instruments are not magic remedies for playing higher. All trumpet players will be able to play the same high note no matter what instrument they are playing.
Haydn Trumpet Concerto on E-flat Trumpet
Stravinsky’s Rite of Spring using a D Trumpet (as it is scored for) instead of the normally used Piccolo Trumpet