I think that as long as his head and back are supported properly, any age is okay to put him in the swing. I know people who put their babies in swings within the first couple of weeks, others wait a while longer. If your baby is comfortable, I wouldn't worry about age.
As long as the babys head is supported and he is secured in the seat, and the swing doesn't jerk him around it's fine. I started using mine right away because it helps them sleep. I have seen swings that have two attachments: a regular seat and a bed attachment. Go with your instincts, thats the best answer.
I also used mine from an early age -- they both loved it! Alot of the newer swings actually recline way back so that they are basically lying down.
I never had one that fancy, so I used to prop their head and necks up with rolled up blankets/burp cloths or with Cody, I bought one of those tiny boppy-looking pillows that fit around his neck to hold it up.