Do mutant alleles result from mutation of the wild type?
No, not necessarily!

The terms might be confusing I agree. The most common allele (the wild type allele) might be a mutant of one other type of allele (one mutant allele) present in the population.

The reason we call the least common alleles the mutant alleles is that beneficial mutations are rare compared to neutral or deleterious mutations. If at a given locus most mutations are deleterious, then the mutant alleles will never reach a high frequency. However, if at a given locus a beneficial mutation occurs, then this beneficial mutation will likely rise quick quickly in frequency and therefore there is only a limited time frame under which the most common allele is actually a mutant of a least common allele. In consequence, we tend to call the least common alleles "mutant alleles". This name is often a good fit to the reality but not necessarily.

Note by the way that there are other sorts of selection regime occurring at a given locus that may maintain polymorphism at relatively high heterozygosity such as any form of balancing selection.