No, Hestu will not always end up in the Lost Woods (or more accurately, in Korok Forest). According to the IGN wiki:

Hestu will finally make it to Korok Forest after you visit him at either Stable.

This answer summarizes Hestu's locations in great detail, though I will summarize them briefly below:

After initially meeting Hestu south of Kakariko Village, and completing the Priceless Maracas Quest, he will then move to a location south of Kakariko village (along the winding path) in west Necluda.

After 2 upgrades here, Hestu then moves to the left bank of the Hylia River, just northeast of Riverside Stable. Sometimes Hestu can also be found at the Woodlands Stable. And some sources suggest he can also sometimes be found at the Wetlands Stable. Nevertheless, more inventory upgrades are now available for purchase.

Finally, Hestu will make it to Korok Forest where he will remain for the rest of the game.

No, Hestu will not always end up in the Lost Woods (or more accurately, in Korok Forest). According to the IGN wiki:

Hestu will finally make it to Korok Forest after you visit him at either Stable.

This answer summarizes Hestu's locations in great detail, though I will summarize them briefly below:

After initially meeting Hestu south of Kakariko Village, and completing the Priceless Maracas Quest, he will then move to a location south of Kakariko village (along the winding path) in west Necluda.

After 2 upgrades here, Hestu then moves to the left bank of the Hylia River, just northeast of Riverside Stable. Sometimes Hestu can also be found at the Woodlands Stable. And some sources suggest he can also sometimes be found at the Wetlands Stable. Nevertheless, more inventory upgrades are now available for purchase.

Finally, Hestu will make it to Korok Forest where he will remain for the rest of the game.