Not necessary, but definitely recommended.

Newer content will not be generated in biomes that you have already explored.

Okay but what is biome explored? I just run through whole uncovered mountain biome, finding four chests and not a single onion seed. Map was partly uncovered as I was on the plains in the same island. Plains also without single tar pit.
So my question is what to do if I really don't start once again? Sail to completely different part of map? Will it help, I feel like I lost few hours today on surching things that are impossible to find.

To clarify, where your character has physically been is explored and about 500m outside those areas is where new content is.

New game is nice to get used to the whole experience and progression as there has been lots of optimisations and tweaks all over the place that you may not notice properly if you don't go looking.

Not like I know how much ist 500 meters, but thank you, answer is actually quite helpful. And I am considering starting over, I am still not sure If I l'll be patient enough to go through many quite hard moments. So likely I would rather test first what is new.
Then I am back to plan A - sailing to some different part of map and there looking for mountains and plains.

Thanks alot guys!

Regarding distance:
Your default explore range is 100 meters. The land on your map appears at that range as you explore. From that you may get an idea of how far 500 meters would be.