H1 transfer with limited time left on current status

[Amstuckusa]

My current H1-B Visa is valid through July 2019 and my company just informed me that they won't be filing for an extension per some changes in their immigration policy. So, I have about 4 months left on my current Visa but have an approved I-140. Does anyone know if a new employer could file for a new h1-b Visa (aka transfer) and then for an extension at the same time (or prior to the July expiration date)?

Also, how does the GC status factor into this?

Thanks in advance for the help.