Council chiefs say anyone who has a headache, aches and pains, sore throat, runny nose, sneezing episodes, a tummy ache can get a test as a precaution.
It includes people who feel very tired for no good reason, with a PCR test being sent off to the lab and people receiving their results within a few days.
The three traditional symptoms of the virus include a new continuous cough, a high temperature and/or a loss or change to a sense of smell or taste.
Dr Lisa McNally, Sandwell’s director of public health, said: "It’s better to be safe than sorry. So please, if you have any of the three traditional symptoms, you must book a test immediately and self-isolate until you get your result. If the test result is positive, you must self-isolate for 10 days.
"And if you’re feeling unwell with any of the other symptoms, I would urge you to book a PCR test and it’s sensible to stay at home until you’ve received your test result. Getting PCR tests and self-isolating when you need to will help break the chains of transmission and can save lives."
People should select the option "I've been told to get a test by my local council, health protection team or healthcare professional" when booking a PCR test, then select "My local council or health protection team has asked me to get a test, even though I do not have symptoms".
Councillor Suzanne Hartwell, cabinet member for adults, social care and health, added: "I urge residents to book a test if they are feeling unwell. It's really easy and only takes a few minutes. Help stop the spread of Covid-19 this autumn in Sandwell."
To book a test and for more details, visit sandwell.gov.uk/testsandwell.