Immerse the canning jars in simmering water until jam is ready. Wash new, unused lids and rings in warm soapy water.

Mix together 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 the pot to 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 the rims with a clean, damp cloth. Top with the lids and screw on the 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 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 the lid does not move up or down and seal is tight. Remove the rings for storage and store in a cool, dark area.