Generally speaking, no. Poison ivy spreading is caused by coming in contact with the rash-causing oil, urushiol.  Once a poison ivy or oak rash appears (i.e., within 1-3 days), the original oil has all bonded to the skin, so it can't be spread to others.

If you experience poison ivy rash spreading after this time, it is most likely you are coming in contact with the oil again and possibly through secondary contamination. It is not uncommon for urushiol to be picked up from the surface of an item that has come in contact with the plant. Urushiol can stay on surfaces such as tools, gloves, boots, or even a pet's fur for quite a long time and a tiny amount can cause a reaction.

The best practice to avoid spreading the rash is to wash the area with a cleanser designed to remove urushiol such as Tecnu Original Outdoor Skin Cleanser. Tecnu Original can also be used to remove urushiol from secondary sources of contamination such as tools, gloves, boots and fur.

For more information, see  "Unexpected Sources of Poison Ivy/Oak Rash"