Many people prefer a value of .8 rather than .6, so something below .6 is indeed likely to be questioned. Have you examined "alpha if item deleted" as an option to find out if any one item is lowering your overall alpha?

Although it is a 'poor' value, it is not 'unacceptable'

Thank you David. Is "alpha if item deleted" available in Stata?

I do not know about Stata, but you can always create your own version by systemically rotating one item at a time out of the full set.