Of course you can take detours, like the Silvretta Hochalpenstrasse (spectacular!) and avoid Arlbergpass. Other options are to go south and really take in the Alps. From Basel to Chur and then go for Lenzerheide, Julierpass, Berninapass, via Livigno to Bormio, Passo Stelvio, Reschenpass and you're back in Innsbruck. That latter option might be a bit ambitious ;-)