A restraining order that does not change the existing status quo is not likely to be stayed, so a retraining order here, invalidating the new laws but leaving the original NY laws in effect, is a good "no stay" case. By contrast, in Wollard, the injuction barred the Md. "may issue" law, and would have resulted in a massive change in the status quo. The trial court refused to stay his ruling, but the Court of Appeals did; and although it set the case for expedited briefing, and the case has been argued, no decision has yet been issued and the stay remains in effect.
That said, i am not sure that a TRO is an appealable order, although perhaps a petition for writ of mandate would lie. The TRO is just the first step--there still has to be a trial and a determination by the court that the law is invalid, resulting in a permanent injuction (which is an appealable order). A TRO here doesn't mean that much, except that the SAFE law will be delayed until the litigation is concluded.