Why doesn't the Byakugan have a second stage?
There is a &quot;second stage&quot; of the Byakugan, called Tenseigan. And it has its own Chakra Mode, which is said to be on par with the Naruto's Nine-Tails Chakra Mode.
When the Byakugan is combined with the chakra of an Ōtsutsuki clan member, it evolves into the Tenseigan. The only known wielders are:

Hamura Ōtsutsuki
Toneri Ōtsutsuki


The Tenseigan has abilities similar to the Rinnegan.

The Tenseigan allows the user to control both attractive and repulsive forces, similar to the Rinnegan. Additionally, it also grants the wielder Tenseigan Chakra Mode, which in turn gives the user an increase in speed, power, strength, durability, and reflexes. It also grants its user Truth-Seeking Balls, thus granting the user all five nature transformations and Yin–Yang Release.
source: Naruto Wikia