Hiccups are little more than a reflex. You get them when the vagus nerve or one of its branches, which runs from the brain to the abdomen, is irritated. Experts say hiccups are most often a reaction to common digestive disturbances. And they're usually more a nuisance than anything else.
While Finasteride is linked to psychological conditions such as depression or suicide, I cannot find any reports of Finasteride causing hiccups. However, we are all individuals and you may be experiencing a side effect unique to you. One way to find out is to stop the Finasteride and if the hiccups go away, that might be the cause. You might want to start and stop one more time to be sure. Next have your doctor give you a trial of Duteraside to see if you can tolerate that drug.
One caution though, adverse reactions are sometimes not reversible. We must weigh the benefits verses the risks of all medications we take.
One way to stifle hiccups is to overwhelm the vagus nerve with another sensation. There are numerous techniques to do this without additional drugs. Just google "how to cure a hiccup" and there are numerous examples. My favorite is to put a spoon in a glass of water and try to drink somewhat bent over. Try not to drown yourself. It sounds stupid, but it worked for me.