With respect to Sieur Zahn, I believe his recollection of his story is incorrect. Unless the story is "The Evidence of Things Not Seen" instead (which I consider less likely), it's almost definitely "Cascade Point."
Captain Pall and his crew have to deal with a navigation error caused by an unaccounted-for piece of Ming-metal that throws off their rotation through Colloton space, causing them to arrive at a destination, which while in the correct place is not the right planet.
Specifically, he is incorrect about people being sedated for the rotation through Colloton space; everybody but the bridge crew habitually take sleeping pills for the transit, not because the time in Collaton space is dangerous, but because it is highly uncomfortable.
Leeds checked in first, reporting officially that the proper number of dosages had been drawn from the sleeper cabinet and were being distributed to the passengers.
Seeing other versions of yourself is an aspect of Colloton space, and somehow seeing different versions of yourself, some very like yourself and some very different is intensely depressing:
Lanton was still alone in the lounge when I arrived. "Doctor," I nodded to him as I sat down in the chair across from his. "How are you feeling?" The question was more for politeness than information; the four empty glasses on the end table beside him and the half-full one in his hand showed how he'd chosen to deal with his depression. I'd learned long ago that crying was easier on the liver.
He managed a weak smile. "Better, Captain; much better. I was starting to think I was the only one left on the ship."
"You're not even the only one awake," I said. "The other passengers will be wandering in shortly—you people get a higher-dose sleeper than the crew takes."
He shook his head. "Lord, but that was weird. No wonder you want everyone to sleep through it. I can't remember the last time I felt this rotten."
The cross of images of alternates is important because it turns out that their motion is repeatable and can be played in reverse.
Then I waited, trying not to think of what was coming... and at the appropriate time I lifted the safety cover and twisted the field generator control knob.
And suddenly there were five of us in the room.
I will never understand how the first person to test the Colloton Drive ever made it past this point. The images silently surrounding me a bare arm's length away were lifesize, lifelike, and—at first glance, anyway—as solid as the panels and chairs they seemed to have displaced. It took a careful look to realize they were actually slightly transparent, like some kind of colored glass, and a little experimentation at that point would show they had less substance than air. They were nothing but ghosts, specters straight out of childhoods scariest stories. Which merely added to the discomfort... because all of them were me.
Five seconds later the second set of images appeared, perfectly aligned with the first. After that they came more and more quickly, as the spacing between them similarly decreased, forming an ever-expanding horizontal cross with me at the center. I watched—forced myself to watch—knew I had to watch—as the lines continued to lengthen, watched until they were so long that I could no longer discern whether any more were being added.
They encounter minor navigation errors, leading up to the discovery that Dr. Lanton's equipment contains a coil of Ming-metal, and then arrive at a completely wrong (barren and uninhabited) version of the planet they were travelling to.
The eventually use Lanton's eidetic recollection of the movement of his images to reverse their rotation, and return to Earth.
After their arrival, and some publicity of their adventure, Captain Pall gets an offer of a position with a liner. And on a subsequent cascade point, there's a new image of him in a liner captain's uniform:
Six months later, on our third point out from Prima, a new image of myself in liner captain's white appeared in my cascade pattern. I looked at it long and hard... and then did something I'd never done before for such an image.
I wished it lots of luck.