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I'm not sure it would be a better/faster method, but you can also take a plane or a boat to Havre-Saint-Pierre which is much closer to Mingan. Sadly, there are no boat crossing directly from GaspΓ©. You have to at least go to Matane or Rimouski. In Rimouski, you could take a boat to Mingan, but it will probably take more time than by car.
One important thing to know, however, is to be prepared. Mingan is not a very popular touristic destination (despite its beauty). Therefore, you should reserve in advance the boat arrangement and your accommodations. At Mingan, you are close to the wild Quebec.