The LED is the most dimming-friendly light source ever made. The problem is that most people are using screw-in "incandescent replacement" LED modules, with obsolete dimming schemes** intended for incandescent lights (which just don't have the refinement to perform well in the low range). This is the ugliest hack in electrical design, and produces ugly-hack results.
If you're such a big fan of dimming, you should investigate better ways to dim. There are several.
LEDs want to dim
A native, raw LED will dim flawlessly, with extremely refined levels of control. Right out of the 'chute, there are two control methods that will work extremely well. One impractical for the casual hobbyist, the other readily accessible.
Dimming by current limiting. Raw LED emitters must be current controlled or they will fry themselves. This is done by a driver circuit that outputs a specific current, varying the voltage to hit that current. LED spec sheets all test and rate them at specific currents. But the spec sheet lists a maximum current -- you can run them at any lower current. And if you do, brightness will be roughly proportional.
A particular LED emitter might have a spec current like 350ma, and a maximum current at 1400ma. If you send it 35ma, you get about 1/10 spec light. At 3.5ma, about 1/100 spec light. 0.35ma, 1/1000. I've driven a 3000ma, 36V array at about a milliamp and gotten a tiny amount of light out of it, just enough to tell what color the emitter was. There's no limit to how dim you can go, provided the driver can go that low, and you have the manufacturer's job of matching driver to LED emitters, heatsinking the emitters, building them into a viable product, etc.
Building a custom current-limiting driver is beyond the light hobbyist, but you might find commercial LED drivers which do this for you, e.g. using a 0-10V dimming system (the type used in commercial lighting).
Dimming by Pulse Width Modulation. A lot of 12V LED products simply use a resistor for current limiting. This involves underrating the LEDs somewhat - they can't run at peak performance - but it also means you can turn the LEDs on and off very, very, very fast. This allows Pulse Width Modulation dimming -- the light is at 50% brightness because it's on 50% of the time. 1% brightness because it's on 1% of the time. The full duty-cycle occurs hundreds of times per second, far too fast to see. PWM dimming is easy for hobbyists using 12V resistor-based products and cheap, off-the-shelf hardware available all over eBay, Amazon, in stores, you name it. Top brands like Leviton are even making modules that "play well with" PWM dimming.
A dimmer or RGB controller makes the pulses, with enough force to directly drive a considerable amount of LED light - like 130 watts (about 700 watts of incandescent). If you need more, gadgets called amplifiers can drive lots more.
So if you don't like how "low" the commercial PWM dimmers/controllers can go -- it's well within a hobbyist's reach to make your own controller using an Arduino or Raspberry Pi. And feed the Adruino's weak output into an off-the-shelf amplifier to control the lights proper.
Better dimming products exist
There are also commercial dimming technologies. One example is the "0-10V" system. In this, the dimmer sends a signal (0-10 volts obviously) on a separate cable to LED driver modules which understand that signal. And then, they can go as dim as the driver design allows. (the LED itself provides no lower limit). So it's just a question of finding a quality fixture or driver that takes 0-10V for dimming.
Smart switches/controls are another way to solve this. In this case, they use either a) communication wires, b) a radio scheme, or c) modulating signals over the power line, to communicate between a smart dimmer and the smart LED lights. It's a digital signal that says exactly where to set the brightness, so in theory, this is a viable way to get any brightness the driver can handle. Of course, this can be botched in implementation, either with a driver that doesn't handle "very dim" well, or with the digital "notches" being chosen too coarsely.
** Obsolete dimming schemes such as leading or trailing phase control, like these ratty old obsolete things listed on this page. Those dimmers are designed with a MANDATE to be backward compatible with incandescent - in fact on