Employer Green Card Sponsorship: When?? -- ANYTIME!
A U.S. employer can sponsor any foreign national for a green card from any U.S. visa status, or directly from abroad
An employer can sponsor someone for a green card from ANY visa status and it is also legally permissible for an employer to sponsor someone from abroad for a green card.
Some people in the US in F-1 student status or one of the treaty-based work visas (TN, E3, H1B1) think that they MUST get H-1B status in order to be sponsored for a green card. That is not true.
There are various things to consider and for people born in India or China, there are special considerations given the slow process but certainly Mexican-born TNs or Chilean-born H1B1s or Australian-born E3s can all be sponsored for green cards even though those visa categories are not 'dual intent' visa categories.