Leave to remain -- new contract starting date

[nikuss]

Hello everyone,

This forum has helped me before (thank you!) and here I am again, trying to plan my next application.

Current status: Tier 2 General entered 19/03/2017 valid until 13/04/2020.

I am a postdoctoral researcher and I am trying to understand if I can apply for a grant in time to make a leave to remain application. My current contract ends 01/04/2020 and a successful grant application would mean a new contract with a new start date. As far as I understand, as long as I apply before 13/04/2020, my leave will be extended by Section 3C. However, what is the allowed range of start dates on the new COS? Does it have to be within 14 days of my previous leave expiring, or can it be full 3 months before I'm employed again?

Sorry if my question sounds convoluted, I am ready to provide more details if needed. Thanks!