Alleviating pressure from coccyx with padded shorts
Biking shorts could help but it's good to understand some things. First is that your coccyx shouldn't be touching the saddle or minimally if it does. A lot of people make the mistake of getting a cushy bike seat to alleviate such pain. The problem with squishy bike seats is that your sit bones sink down into the saddle and all the squishy stuff packs into your crack and starts pushing on stuff up there. Imagine a really hard seat - your sit bones contact the saddle and nothing squeezes up into the soft area. Of course, between the bones and this hypothetical hard saddle is tissue and skin which can get irritated and bruised. Which is why we have padded saddles. Appropriately padded saddles.

So the first thing I'd suggest is trying a new saddle. Something firmer and made out of quality materials. Stock saddles can be pretty bad. 

The chamois in bike shorts is like a second pair of socks on a hike. It adds a layer of padding and friction relief in that bones/tissue/saddle interface.

In short, bike shorts might help but the right saddle alone will enable you to comfortably complete an ~11 mile ride without butt bruises.

Caveat - exceptions to all of this if you are anatomically abnormal in any way, I am particularly thinking of overweight.

Caveat no. 2 - it takes time to get used to being back on the bike. Your butt is going to scream at you if you over-do it and cause some tissue bruising. And if you do that you aren't going to want to get back on the bike. So take it easy!

And oh yeah - look at saddles with anatomical cut outs, those work really well for some people. 

And oh oh yeah. Some stores have a test program where they mail you like ten saddles and you ride all of them as you like in order to find one that fits. http://www.sportfit-lab.com/bike-seat-demo.html. WTB also has a ten-pack that they sell shops - shops that have that should rent you a saddle for a week for like 10 bucks.