Line an 8-inch baking pan with cooking spray and parchment paper, leaving an overhang on two sides; set aside.
In a large pot, bring apple cider, ginger, cinnamon sticks, and the peel of the orange to a simmer (save the rest of the orange for another use).
Remove pot from heat; cover with plastic wrap and set aside to infuse for 30 minutes.
Whisk to combine 2 ½ cups sugar and pectin in a bowl; set aside.
Grate the apple (unpeeled) using the large holes of a box grater. Discard the core. Set aside.
Use a slotted spoon to remove orange and cinnamon sticks from cider mixture. Bring back to a simmer.
Add corn syrup and sugar mixture to cider. Stir to dissolve. Add grated apple and stir.
Cook over medium-high heat, stirring occasionally, until mixture registers 235°F on an instant-read or candy thermometer, about 40 to 45 minutes. (Note: If you don't have a thermometer to use, check the consistency by placing a spoonful of the mixture onto a chilled plate. It should set quickly. When it's firm enough to hold its shape, the candy is ready.)
Remove pot from heat; stir in salt and rum, if using. Pour into prepared baking pan. Set aside, uncovered, at least 4 hours or overnight.
Lift pâte de fruit from pan (it should release easily). Flip onto a cutting board; remove parchment.
Cut into 6 rows, cleaning your knife between cuts. Cut each row into 8 pieces.
In a medium bowl, stir to combine remaining ¼ cup sugar with cinnamon. Toss a few pieces of pâte de fruit at a time into the sugar mixture to coat. Transfer to a baking sheet or serving platter.
Store candy in an airtight container in a cool spot for up to two weeks.