Adding another better solution: This displays only the first image in a gallery, linked to launch the carousel. You would make that first image, and it’s caption, clearly indicate it as a link to a gallery.
Give gallery an HTML ID:
<div style="display:none">
</div>
Then add CSS:
#gallery-link .tiled-gallery-item { display: none; }
#gallery-link .tiled-gallery-item:first-child { display: block; }
Note: .tiled-gallery-item is the class inserted by Jetpack when Settings> Media> Image Gallery Carousel is set to Tiled Galleries.
Here’s old original answer, which has some value, and works fine when launching a gallery on another post, but not on same post:
This is tested, but NOT rigorously; seems to work: write the shortcode with the images you want (attachment IDs); put it in post, but don’t display it:
<div style="display:none">
</div>
Then you can trigger the Carousel w/ a href URL that looks like:
http://examples.com/your-post#jp-carousel-111
where “111” is the ID of the first image attachment ID you want Carousel to start with.