What's wrong with what you have? What are you trying to change or improve? Your headphones are highly regarded.
All headphones and speakers sound
different (better or worse) but most electronics are better than human hearing unless you overdrive and amplifier (or something) into clipping/distortion.
It's not "hard" or expensive to build a good headphone amp. You usually only need less than 1 Watt and you certainly don't need 1950s tube technology....

Most of the cost of a headphone amp goes into the case, connectors, controls, power supply, etc. And it's a "specialty item" so the marketing & distribution costs are high compared to high-volume products like TVs & AVRs. The actual amplifier electronics are cheap (at least with modern solid state electronics).
If your headphones aren't going loud enough you need an amplifier with more output voltage. (The HD650 is higher-than-average impedance and that generally means less sensitivity.
According to the review here, your headphones are
slightly lacking in bass so simply boosting the bass may be the "solution". And boosting the bass may also require more output from the headphone amp.
I’m new to the audiophile world
Beware!!! MOST of the audiophile community is nuts!

This is one of the few scientific-rational resources. Most "audiophiles" and reviewers don't trust measurements and "don't believe in" blind listening tests. They are also often fascinated with tubes, vinyl records, and other outdated technology. And, a higher price in the audiophile market tends to make it more desirable, even if most can't afford it!
Amir (our host here who does most of the reviews) doesn't do blind listening tests (it's VERY time consuming to do it properly and not everybody has the same hearing ability) but he backs-up what he hears with measurements.
See
Audiophoolery. He talks about the few REAL characteristics of sound quality and it will help you to spot (and hopefully ignore) all of the nonsense terminology about the "sound" of different amplifiers, etc. Also see
What is a blind ABX test?
The components are very ordinary (for example, the tubes are just regular types) and the manual assembly — how could that possibly be better than other devices that are assembled by machines and professional technicians with higher-end components? I’m sorry if this is a silly question
It's not a silly question, especially with all of the "audiophile nonsense" you might be reading...
IMO - tubes are dumb.

There is no technical advantage.*
You can make a good amplifier with tubes OR solid state electronics. But it's more expensive to to make one with tubes. Tube power amps require an output transformer to drive the low impedance of speakers, and that adds even more cost and difficulty. Impedance can be an issue with tube headphone amps too, but your headphones are high-enough impedance that it shouldn't be a problem.
The tubes don't have to be "special", just appropriate for the application. The important thing is a "good" circuit design. Tubes, transistors, MOSFETs and IC's all have tolerances and variations and the engineer's job is to make the amplifier work properly as long as the parts are in-spec. Some tube amps are NOT well designed so you can get variations swapping tubes, or as the tubes age, etc.
* Guitar players tend to favor tube amps, especially for for the way they sound when over-driven into distortion. (You're not supposed to over-drive your hi-fi amp.) And guitar amps are not intended to be high-fidelity. You wouldn't want to hear the whole band through a distorted guitar amp! And they don't all sound the same so most guitar players have their favorite guitar and favorite amp. Of course, they make solid state guitar amps that distort like tubes, and software to simulate the sound of different amps & cabinets, but most musicians are passionate about their instrument and equipment.
