Adding an additional tenant to an AST

Tenancy Types and Management

I took on a (UK resident) tenant in November 2022 and it was agreed between ourselves through an addendum to his AST contract that when his partner and young child (non-UK residents) received their UK visas to join him in the UK that the contract would be amended to add them on as long as all right-to-rent and reference checks were passed.

The addendum was written as such:

"Main Terms of the Tenancy Agreement

Tenant
We, the Landlords, upon receipt of all passed Right to Rent checks and References as set out in Section 22 of the Immigration Act 2014, will accept (partner name), wife of (tenant), to be added to the Tenancy Agreement as a joint Tenant and will amend the initial contract to reflect this.

Permitted Occupiers
We, the Landlords, will accept (underage child), son of (tenant) and (tenants partner), to be added to the Tenancy Agreement as a Permitted Occupier and will amend the initial contract to reflect this."

My query is, do I need to make a whole new AST contract with their names and details added or do I just add additional addendums to the original contract with the details? My thinking is that I create an addendum to say something along the lines of "as of day/month/year, the original AST contract has been amended to now add partner and child to it" and then just add their details where required on the contract.

However, if that is not correct, if I need to make a new contract, would I change the agreement start date, the initial fixed term time etc to reflect the partner and child now being added?

Thank you for your help!