The main walk from Skagfjörðsskáli starts at the next bay east from the hut at Slyppugil, a gradually widening, wooded gully which you follow uphill to the jagged east–west ridge of Tindfjöll. The trail then weaves along Tindfjöll’s gravel, landslip-prone north face to the solitary spire of Tröllakirkja, before emerging onto open heath at Tindfjöll’s eastern end: the double-tipped cone 2km northeast is Rjúpnafell, which can be climbed up a steep, zigzag path to its 824m summit – give yourself at least five hours for the return hike from the hut. Alternatively, cross the heath to descend back into the valley down Tindfjöll’s southern slopes via Stóriendi; the path is intermittently pegged and often seems to be leading off the edge, but always reappears, with some fantastic views of Þórsmörk’s eastern end along the way. You end up down at river level approximately 2km from your starting point; this circuit takes around four hours.
The most obvious local excursion from Básar is straight up Réttarfell, the peak above, via a low saddle – the path is easy to follow though tiring. The next peak east of Réttarfell, and connected by a walkable 5km ridge, is 805m Útigönguhöfði. Views from the top are stupendous but be warned, the final section is incredibly steep, with chains to help you. Give yourself four hours from Básar for the round trip.
One final option is to hike from Þórsmörk to Skógar in around ten hours, via the site of the 2010 eruption at Fimmvörðuháls.