Heat five 12-ounce jars in simmering water until ready for use. Wash lids and rings in warm soapy water.
Mix peaches, sugar, lemon juice, cinnamon, nutmeg, and allspice in a large pot. Bring to a boil; cook, stirring occasionally, until peaches are soft, about 15 minutes. Remove from heat.
Mash peaches with an immersion blender or potato masher to desired size and texture. Return to the heat; continue cooking jam until thickened, about 10 minutes more.
Pack jam into hot jars, filling to within 1/4 inch of the top. Wipe rims with a clean, damp cloth. Top with lids and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower in jars using a holder, placing them 2 inches apart. Pour in more boiling water to cover the jars by at least 1 inch. Bring the water to a rolling boil, cover the pot, and process for 10 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool, about 12 hours. Press the top of each lid with a finger, ensuring that lid does not move up or down and seal is tight.