Am not sure if Donald Reinhardt wrote this in his books but it was a common saying he gave to his students interested in connecting and mastering their upper registers.
"Don't fall in love with your lower register sound"
It's a good piece of advice. And while I can't speak specifically to Reinhardt's intent I can explain what it means to me. So please excuse if I don't perfectly describe Dr. Reinhardt's methodology. Going to mention only what it means to me.
At the beginning of learning how to play high notes some 41 years ago I was one of those students who needed to hit the High F's loud at first. Early on they didn't come out softly very well. In fact the reason why some cats find their high ones loud at first would make an interesting thread all by itself. Anyway it wasn't until becoming a college music major and studying the Clarke Studies SOFTLY that I began to gain the understanding of the value of playing softly in all registers.
By taking the volume and sound down a notch or two it makes register connection much easier. Conversely when playing the middle and lower registers loudly it tends to leave the mouth corners too loose and unfocused. That and there is the marked tendency to drop the jaw. Or recede the jaw/teeth unfavorably in those persons who might be better advised to keep their jaw forward. I play with a receded lower jaw myself just fine so it doesn't matter to me specifically if I "drop my jaw" but this advice is a variable matter in each player.
I also play with another chop setting which requires a forward jaw. So as such I can see the wisdom of not dropping the jaw in certain applications.
Another advantage of softer playing in the middle, lower, and upper registers is that it is a great way to reduce arm pressure on the chops. One of the more interesting discoveries I made some five or so years ago was that arm pressure doesn't correlate with high note production. You can prove this to yourself by playing a note well below your highest limit and then adding only arm pressure. Even if you blow a little harder the tone won't rise very much. A major second is about average. Try it? Play a High C, blow a tad harder while adding more arm pressure but apply no muscular contraction to your embouchure.
Guess what? If you're average The note won't rise above the D natural. Instead the rule is that arm pressure correlates with volume production NOT register increase.
And within these words lies a clue: By learning to play your upper register softly you can remove excessive arm pressure from much of the game. Thus this kind of development lets you play a High E with only normal placement arm pressure. About the same pressure that gravity gives your horn when it is placed upon the palm of your hand.
See Roy Roman's video here: Roy Roman plays a double c with no pressure - YouTube
Hint: You don't need to work your chops up to a Double C as Roy does. He plays a specific chop setting known as the "Stevens-Costello Triple C Embouchure". Upon this setting it is fairly easy to blow with minimal pressure even well above Double C. Most trumpet players however will not be able to make the conversion to Stevens-Costello. I use the video only as a reference on how to lay the horn on your palm to attain only slight contact pressure...
For most of us "human" trumpet players (ie not especially gifted with the ability to play settings like Stevens-Costello) we can still derive much valuable help from laying the horn on palm and playing with the mere contact mouthpiece pressure allowed by gravity alone. In time you can take that squeaky High C and turn it into a mezzo forte C. Later a forte and then double forte.
I'm not particularly gifted naturally in the upper register. Meaning that like most of us I can't play Stevens-Costello as described in the system. However I was still able to build a decent High F and G at least at a mezzo forte volume with just the horn resting on the palm. If I want a little more volume? I can crank the pressure up a bit. Keeping everything within acceptable levels which won't blow my chops out in the short or long run. And of course if it is the last set of the evening and a few High F's or G's are necessary? Well at that point I may jam it good and hard. However this isn't my whole way of playing like a lot of cats blow.
Some lead players will use heavy arm pressure all the time when blowing above the High C. Maybe even notes as low as G top of the staff. For these type pressure is a way of life. "Five minute heroes" a humorous term to describe them.
So by all means experiment with keeping the volume down in the middle and lower registers before you ascend above the staff. Concentrate on minimal chop muscle motion too. Use the same setting from low to high and you''ll soon find that High E with only minimal arm pressure. And at that point its pretty much impossible to run out of chops on a gig. At least up to the High F to G or so. And if you have that ability?
You're definitely among the minority on this forum. Very few trumpet players who post here can play a solid two to three sets of a standard lead trumpet gig. Most of them won't admit this is so but you can bet money on this being true.
Caution: Until playing with less arm pressure above the staff gets well grooved into your ability don't trip too much on reducing pressure while performing. Just concentrate on the music. In some ways the above advice may seem to contradict my suggestion to preach the "Just tongue and BLOW" simple ideology. Not really though. "Just tongue and blow" is a very valuable way to develop good air support and a positive, aggressive, confident lead performance style.
In short use as much arm pressure as you need to fulfill your required music. Use your head though. If you really can't play thirty High C's (or similar) on a gig? Take the notes down an octave or delegate them to the cat playing second or third etc. He needs the experience too.
But in practice and in most band classes? Don't use undue arm pressure very much. You're just "rehearsing your mistakes". Play with your head, not the ego. As Maynard said:
"Your own hands are directly connected to your ego". A reference to using excessive arm pressure on a gig.
In fact I will post that video too. The quote is at about 1:11 here: Maynard Ferguson Clinic: 05. Upper Register - YouTube
And have fun. Enjoy yourself!