John Steakley died a few years back, apparently while he was working on a sequel to Armor, but as far as I know it wasn't complete enough that it'll ever be released. The only other things I've seen from him were two stories set in the Horseclans setting which weren't bad, but not nearly as good as Armor or Vampire$.
I'm trying to think of other stories like The Private, but it's not easy to come up with something that's exactly the same. William C. Dietz has a series that starts with Legion Of The Damned that is basically The French Foreign Legion INNNN SPAAAAACE! and includes cyborg soldiers who are human minds implanted into robotic bodies. The process is only done to humans who are about to die, and one of the viewpoint characters is a guy who was sentenced to death and given to the Legion after he died. That's the only "condemned criminal goes to war" equivalent I can think of off the top of my head.
Another military SF series that I enjoy quite a bit is by Elliott Kay. It starts with Poor Man's Fight. The main character is a kid who just graduated from high school and is forced by exorbitant student loans to go into the military. So he's not a criminal, but the bit about not really wanting to be there is present. The story follows him through training and into his first deployment where he finds out that he's not really a fan of the military but he's very, very good at violence and thinking on his feet. It's a great adventure yarn and I highly recommend it. The sequels really open up the setting and expand the story.
Old Man's War by John Scalzi is also a great read, and also the first in a series. Earth is pretty badly isolated from the rest of the galaxy, and the only way first world citizens ever get into space is by enlisting in the armed forces. On their 75th birthday. For whatever reason, the military that protects human space only accepts senior citizens, and the main character enlists. We follow him through training and out into the wider galaxy, where he encounters a variety of aliens who are at least as odd as the lizards in The Private.
Speaking of lizards, you might also like The Bug Wars by Robert Asprin. There are no humans in the book, it's told from the point of view of lizard soldiers who are fighting against an insectoid enemy. The lizards aren't very much like the ones in The Private, but they're very interesting just the same. I don't think the book is in print anymore, but Baen books makes it available as an ebook.