No, if you fail a morale test from fear you only suffer the consequence written in the fear rule description. That being said, I feel like this needs to be re-written to avoid confusion.
You could just call it a Fear Test. Given that a Morale Test is taking a Quality Test and then dealing with the results, a Fear Test is just taking a Quality Test and then dealing with the (different) results.
I like my tea, as I like my nights: dark, endless, and impossible through which to sleep.