If hot packs do not work, try an ice pack. Sometimes, engorged veins need a cool pack to contract and then the milk duct which was constricted due to engorged blood vessles, gets the space to relax and milk begins to flow out of that duct.
5 mins for a ice pack is what my lactation expert told me.
hot bath makes you feel good.
Heat packs, massage it, if you also breast feed try nursing in different positions.. If it starts getting warm and red call your dr asap in case its developing into mastitis.
I had this a couple weeks ago and it was quite painful. It lasted about two days. I kept feeding from that side and massaging it while pumping or nursing. It worked itself out.
Sounds like you might have a clogged milk duct. Try a really hot bath or shower and hand express it. It might take more than one try especially with the pain. If things persist call your doc and they can maybe let you know more on what to do!!