We can sponsor H-1B transfers, TN visas for Canada/Mexico citizens (we did this for Render's first engineer), H-1B1 visas for Singapore citizens, and pretty much any other work visa that's not a new H-1B. If you're on an OPT, we will file a new H-1B for you because you're already here.
We will eventually grow out of this constraint, but it's personally frustrating because my own first job out of college was on an H-1B visa.
I thought H-1B was the only way to immigrate to the US for work? There is L1 but L1 requires the employee to be working for the foreign office of the US company for at least a year before he/she can be issued L1.
So which are the visas you are sponsoring when you say "all visas except new H-1Bs"?