Each species of flowering plant produces a set number of floral parts, more or less. Having three styles and three stigmas per flower is a trait that is determined genetically. Mutations can cause floral parts to be duplicated. Heirloom tomatoes have mutations in the fascicated and locule-number genes, resulting in more locules being produced per fruit. Mutations in the fascicated gene also cause increases in other floral parts, such as petals, sepals, and stamens. If you compare a regular tomato to an heirloom, you'll notice that heirlooms tend to have around 10 petals and sepals compared to a regular tomato's 5.
Now the question is, do such mutations exist for Crocus sativus? A quick Google search didn't turn up anything other than the normal three-stigma flowers, so it doesn't look like such a mutation is common, if it exists at all. My answer to your question is: there is no easy way to get a saffron flower to produce more stigmas, for if such a method existed, everyone would be using it. I don't doubt that a dedicated botanist could come up with something (crossing with a related species, perhaps), but it's not something you can likely accomplish in your home garden.
User Sleepses found an article where a high-yielding mutant with 5 stigmas was obtained through gamma radiation mutagenesis. I'm not sure whether or not this mutant is for sale commercially, though.