That's right, H-1Bs for certain nonprofits (cap-exempt entities) are not subject to the annual lottery/quota but if one wants to move from a cap-exempt H-1B to

That's right, H-1Bs for certain nonprofits (cap-exempt entities) are not subject to the annual lottery/quota but if one wants to move from a cap-exempt H-1B to an H-1B with a cap-subject company, then one needs to go through the lottery. However, part-time concurrent H-1B employment with a cap-subject company is allowed. There is no restriction on pursuing a green card if one is working for a cap-exempt entity.