Add Additional place of business.

You have to take separate gst registration in Tamilnadu

GST registration is granted state-wide so you have to apply for a new registration in Tamil Nadu.

GST is based on state. Additional place of business means a place within the state itself. If you want to register in other state you have to register separately.

U have to take seperate gstn number for Tamil nadu .

As per provision of GST Act, A registered entity would have one GSTIN per State, that means a business entity having its branches in multiple States will have to take separate State wise registration for the branches in different States.

But within a State an entity with different
branches would have single registration wherein it can declare one place as principal place of business and other branches as additional place of business.
However, a business entity having separate business verticals (as defined in section 2 (18) of the CGST Act, 2017)or separate place of business in a state may obtain separate registration for each of its business verticals/business places.

So, in your case you are required to take State wise different registration.