Anyone with a glancing knowledge of theory will tell you that of course there are reasons why SUSY and strings should be considered. Is the SM complete now that the Higgs has is all likelihood been detected? It could be argued. Does the SM still have significant problems? Yes.
One obvious problem is that there is no gravity. But even if this is not a problem for you (maybe you no like TOEs?) there are still theoretical issues. Some have been expanded upon in this thread. For one thing, it seems that SUSY is probably needed in order to achieve gauge coupling unification. But maybe you no like GUTs either?
Yes the SM describes low-energy stuff that we're used to very well, but there are still some theoretical issues and, to be honest, it's a bit of a messy theory. If you desire any kind of mathematical eloquence in your theories of nature, the gauge group U(1)xSU(2)xSU(3) might strike you as a bit ugly. And all those parameters! All need to be fine tuned and are not predicted by the theory.
No, if you desire beauty and completeness and a theory free from errors, you're definitely looking at physics beyond the SM.