I got offered the job but I don’t know how to go about handing my notice in?

I live in the UK. I got offered a job today that I worked really hard for and actually had a two hour interview for yesterday. Better pay, better hours and more opportunities for progression.

I went in after work to fill out some paperwork for HR. The assistant manager told me that HR would be in touch to discuss references etc, in the next two days. I have a start date of November 1st and have to give 4 weeks notice at my current place.

Would it be too soon to hand in my notice at my current job when I haven’t signed a contract yet?

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