The town's infection rate reached 94 per 100,000 on Friday, up from 64 a week ago.
Analysis shows cases are currently most common in those aged 20-39. Of 133 positive cases in the seven days to June 23, 77% were aged under 40.
Mayor Preston said: "Our rate is fast approaching 100 for the first time in months.
"We're seeing more disruption to families through isolation with crucial school time missed and adults required to stay off work.
"We've got to keep the numbers as low as we can in the hope our local NHS services don't come under strain again.
"Please follow the rules. The end is very much in sight, but we need people to get both their jabs when invited. Two doses are vital - the vaccination programme protects lives and livelihoods."
Cases in the 0-19 age bracket have levelled off, data shows, while cases in the 40-59s have started increasing.
Middlesbrough GP practices are offering walk-in vaccination slots for over-40s at North Ormesby Health Village on Saturday.
Anyone over 40 who hasn't had their first jab or those who had their first AstraZeneca dose at least eight weeks ago can attend between 9am and 1pm.
More drop-in sessions take place on Monday and Tuesday (June 28 and 29). You can find out more via www.borovaccines.co.uk.
Those aged under 40 can now book their slots using the national booking system via www.nhs.uk/coronavirus.
Anyone with symptoms of Covid-19 - a new, continuous cough, a high temperature or a change or loss to the senses of taste or smell - should isolate along with those they live with and book a test. Tests can be booked online via www.nhs.uk/coronavirus or by calling 119.
All adults can access free rapid tests for those without symptoms. Details can be found via www.middlesbrough.gov.uk/lfd-test.
