Denise Birdsall, aged 52, left her home in Garforth at about 9.30am on Monday and caught a train from Leeds to the North Yorkshire seaside town.
She was seen on CCTV in Scarborough that afternoon buying a skirt from an Age UK charity shop, which she changed into.
At about 8pm yesterday police were contacted by a member of the public who had found clothes on the beach at Cayton Bay that were identified as hers.
Officers from Leeds District CID are continuing to work with colleagues in North Yorkshire Police and the Coastguard, which is searching the area.
Detective Inspector Paul Hobson, of Leeds District CID, said: “We are continuing to support Denise’s family who have been made aware of her clothing being found at Cayton Bay.
“We are still treating her as a missing person and are keen to hear from anyone who has seen her at any point since Monday afternoon when she was in Scarborough town centre.”
Anyone with information is asked to contact Leeds District CID via 101 quoting log number 585 of July 16.