I have been to Snaefellsnes as well and it its absolutley worth visiting.
I can list some of the other places i've been.
Reykjadalur Thermal Bathing Pools
This is in Hveragerði which is not too far from Reyjkavik. I went in the summer and the weather was kind of atrocious as we made our nearly two hour hike getting drenched in the cold rain. The views while making the hike though were amazing and if we had clear blue skies it probably would be GOAT level scenery. There was some sheeps up there as well.
Anyway when you final reach the destination you can heal all wounds in these glorious thermal pools and never want to leave.
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Just keep in mind that because of the unpredictable weather, you may want to keep your clothes in a backpack or something because there is no shelter from the rain up here whatsoever. Also if u need to get naked or something there isnt any private place to change. Icelanders though do not care about these kinds of things and have no shame about their nakedness.
It was an amazing day and getting back down is obviously a breeze compared to getting up there. Always prepare to have clothes in case of horrible stupid cold rain and winds cos it can happen at a moments notice.
Even if you go in august youll need winter clothes as it will rarely get above 60 out there