If adopted, the new bill would go into effect on July 1, 2014.
Goode has been working since last year to modify the 10-year abatement, but he linked the effort to school funding when he introduced the second bill.
Last week, education activists packed Council chambers and chanted, "Save our schools" as Goode spoke of the bills.
Developers and home builders have credited the tax abatement with sparking the building boom of the previous decade, saying the incentive offsets the higher cost of labor and construction in the city.
Goode said there has been no analysis showing the current abatement is needed to make individual projects feasible.
"Every project does not need the current tax abatement, but every school absolutely needs more money," he said. "So why take money from needy schools for the tax abatement? In the end, who has the greater need? The kids do."
The original bill would have capped the amount of value that could be abated at $500,000 and phased out the abatement during the final five years. It would go into effect in 2016.
The original bill passed out of Council's Whole Committee in June by a 9-7 vote, with one member absent. Assuming the same nine members still supported the original legislation, Goode could have gotten the bill through final passage.
Mayor Nutter, however, opposed the legislation, and Goode would have needed 12 votes to override a veto.
Goode said he moved to withdraw that bill Thursday because he's "simply in favor of the second bill at this point."
"Rather than cause confusion over both bills," he said, "I want to squarely focus on getting the schools money as quickly as possible."
Contact Troy Graham at 215-854-2730 or firstname.lastname@example.org, or follow on Twitter @troyjgraham.