Tier-2G - change of employer and start date questions

[waiwai]

Hi

My new employer is going to issue me a CoS with start date of 2nd of Jan 2020.
However there is a chance that I might delay the joining date and join a week later, which I understand I can do (with permission from new employer) as it will still be within the 28 days of CoS start date.

However, can I continue work for my current employer after 2nd Jan for a week under notice period? My current visa (sponsored by my current employer) is valid till July 2020.