At this time, due to COVID-19, I think you should file a part-year New Jersey and part-year Pennsylvania income tax return, even though you took steps in September to establish residency in South Carolina.
Pennsylvania says if an employee is working from home temporarily due to the COVID-19 pandemic, the Department of Revenue does not consider that as a change to the sourcing of the employee’s compensation. In other words, you remain a Pennsylvania-based employee because you were before the pandemic (though working from NJ).
Your “tax home” is still PA due to your apartment so I don’t think you need to file a SC return at this point.
The IRS says, “generally your tax home is the entire city or general area where your main place of business or work is located, regardless of where you maintain your family home. For example, you live with your family in Chicago but work in Milwaukee where you stay in a hotel and eat in restaurants. You return to Chicago every weekend. You may not deduct any of your travel, meals or lodging in Milwaukee because that's your tax home. Your travel on weekends to your family home in Chicago isn't for your work, so these expenses are also not deductible. If you regularly work in more than one place, your tax home is the general area where your main place of business or work is located.
Once the pandemic emergency is over, you should talk with your payroll department. At that point, you may be a South Carolina resident.
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