I need fill in my front yard; should I wait until spring ?

After New Orleans sewerage & water board ripped up my street for construction a year ago, the verge between the street and sidewalk was never filled enough before sod was placed making this area a good 3-4 inches lower than it should be.
After the recent hurricane, we had to urgently park on the front lawn due to rising water and now there are also large divots in the grass because of this.

So I will need fill for the front lawn, my question is should I spread the fill now (and risk the grass not sufficiently growing) or wait until February when the St Aug grass will be in a growth phase?

Also looks like river sand may not be the best option, is regular sand preferred?