Kelly Jefferies, 42, and her son Knox have not been seen since July 28.
The pair left their home in Patchway, South Gloucestershire, and police believe they travelled north to an area near Manchester.
Officers are concerned for their welfare and have renewed an appeal to find the pair.
Avon and Somerset Police said Ms Jefferies and her son could be anywhere in the country by now - or even living abroad.
Detective Chief Inspector Kristina Windsor said: “Kelly and Knox’s welfare is our primary concern and we’re working extremely hard to locate them to check they are both ok.
“We understand Kelly may not want to be found but we can’t in good conscience stop looking for her and Knox when we’re worried about them.
“We’ve previously issued a photo of Kelly which shows her with her natural brown hair colour but we believe she may have recently changed her appearance.
“She’s previously dyed her hair blonde and so we’re releasing another picture of her today which was taken some years ago to show what she may look like now if she’s coloured her hair again.
“As Kelly and Knox could be anywhere, we’re asking for the public’s help in sharing our appeal on social media as we know this increases the likelihood of them being found."
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