What’s the best way to get to many glacier hotel from the west side entrance (we are staying in Coram)?

Want to hike Grinnell glacier, maps tells us we need to go 122 miles (2 hours) outside of the park to enter through the many glacier entrance. It looks like you could take GTTSR all the way to the Mary Visitor Center, then exit and enter at the Many Glacier entrance. What would you guys recommend? Also, I’m planning on going October 1st, what time do you think I should be there by for parking, is it still an issue this late in the season? Thank you so much in advance for any help.😊