The quest for tone can take you to many places, and when it comes to overdrive, the ideal tone may reside in a specific pedal, amplifier, or a combination of both. Experimentation is key, as is knowing the limitations of your equipment.
Many guitarists set their amp for a clean tone and use a pedal of choice to send the clean channel into overdrive. This simple, practical setup lets you kick on the grit of your choice when the song calls for it. This method lets you use your amp as a clean palette to sculpt your overdrive tone. While straightforward, this setup is not without its setbacks. Your clean channel and overdrive will share the same EQ settings, which may not be a big deal if your pedal has its own EQ controls. However, many popular overdrive pedals do not feature the same EQ versatility as amps (for instance, they may have only gain, volume, and tone controls) so the overall tweak ability between clean and dirty is somewhat limited. If you like the baked-in tone of the pedal a lot, then this setup may work well for you. In many multiple channel amplifiers the clean channel and the lead or dirty channel have very different gain stages, internal tone shaping and even EQ control circuits to produce the intended clean or dirty tone. Instead of driving your clean channel with your pedals, try using your dirty channel with the gain set a little lower. This will give you the better tone shaping and EQ controls for dirty sounds and a little more headroom to handle your pedal for more dynamics. Often pedals are tried with the same settings you would use to get the same level of dirty tone without the pedal and adding the pedal creates mud, so you switch to the clean channel. Turn down your gain a little and try it again. Then you still have a clean channel to use and EQ for clean playing. The added bonus is a third slightly lower distortion with your dirty channel only.
Another popular way to achieve a rich, gritty overdrive is to crank the overdrive channel of a tube amp. This really gets your tubes working, which results in what many consider to be a more organic, dynamic distortion than what pedals can provide. However, many tube amps, especially high wattage amps, require more volume to reach the desired level of saturation than is practical. For this reason, many amps, like the Carvin Audio V3, have switchable wattage, so you can get a crunchy tone at a much lower volume.
Most modern dense distortion sounds are pre master preamp gain distortions. With pedals, these overdrives are not as smooth as an amp that is designed to be a hi-gain amp. This smoothness is usually from the high voltages and signal swing of the amplifiers as opposed to the pedal creating more dynamic and tone in the signal. Here you just have to try different combinations. Again lowering the input gain on the amp a little and using the pedal for the solo sustain edge can create more dynamics. The Carvin Audio V3, V3m and Legacy have two lead channels that are identical, so you can create a rhythm tone and a high gain and sustaining lead tone to switch between.
Side note: Remember when playing in the band you are just one of the instruments putting out sound. If you tweak your tone at home alone, you maybe unknowingly also filling for the rest of the band. A good solid tone needs to cut through the band, but it also needs to let the other sounds of the band be heard. With loads of distortion you will be a small non-distinctive buzz in the overall sound of the band, and with really fat low and high tone you will sonically fight with the rest of the band.
In many cases, it may be the exclusive use of a particular tube amp or pedal that will get you the perfect overdriven guitar sound that’s in your head. Killer tone can be the result of combining an amplifier’s natural tube breakup with a hot signal from an overdrive or distortion pedal, which can also act as a preamp or clean boost to further push and shape the overall tone. There are many shades in between clean and dirt to be explored. Feel free to take your time and experiment with all the different options available. If it sounds good to you, it is good!