... replacing present-tense tread with trod ... is the only example I can think of where a new non-past form has been based on an irregular past-tense form (though there probably are others that aren't occurring to me right now).
To her example I would like to add shodding.
Shod is the irregular past participle of the verb shoe, meaning to furnish with shoes or to cover with a metal guard to protect against wear. Like tread/trod(treaded)/trodden, shoe/shod/shodden is also a relatively low-frequency irregular verb where there may have been enough disassociation in the minds of the speakers between the present- and past-tense stems to facilitate the creation of the new non-past form shodding from the irregular past tense. Or perhaps the etymology is altogether different. At any rate, examples of shodding abound. Summoning Google yields the following.
From a business wire:
Harry's London is pleased to celebrate the 75th annual Academy Awards(R) by shodding several of Hollywood's leading men for the event's red carpet, as well as for the parties that surround this very special event.
From a fiction short story at the Oyster Boy Review:
My father figured that fixing feet would be better than shodding them, so like the kid who becomes a doctor to go one step further than his pharmacist father, the old man graduated from the Ohio College of Chiropody.
From Making the Shoes Fit, an article published at The Desert Sun online:
Although it's still early, the Clarkes have been up for several hours trimming the constantly-growing feet that horses are equipped with and shodding them for extra protection and better traction.