This website can tell you the available shipping options for USPS packages based on the sender and destination:

http://postcalc.usps.com/

In Magento, under Stores / Configuration / Shipping Methods / USPS / Allowed Methods, if only Standard Post is an allowed USPS option:

 


When calculating the price for my Zip, post is an option:

 


However running the same weight package for a new Jersey zip for example, post is no longer an option:

 


The reason for this appears to be a difference how Standard Post is defined depending on the postal region, and for those regions its not available the items must be required to be shipped ground only:

 


Standard Post
Standard Post (formerly Parcel Post) is primarily intended for single-piece mailings which consist of any mailable matter that is not required to be mailed as First-Class Mail and includes USPS Tracking at no additional charge. It is commonly used for gifts and merchandise. There are no bulk prices for Standard Post.
Standard Post is generally presented at a USPS retail service counter where USPS Tracking service can be initiated.
Standard Post prices are based on the zone (distance) to which the parcel is addressed and the weight of the parcel. While any mailable matter not required to be mailed as First-Class Mail is permitted to Zones 5-8, Zones 1-4 items are limited to mailable hazardous materials, live animals, and other “surface-only” items.

 

 

While there is a way in Magento to make these items appear as ground only, you may run into some legal issues with the Postal Service.

All that should be necessary to make USPS options available for these other zip codes is to select which available USPS options you are able to ship out.

 

You may want to go through the list and select any other options you may want, then be sure to click save in the top right.

Again these will be under Stores / Configuration / Shipping Methods / USPS / Allowed Methods.

Then, the fastest way to see if it is an allowed method will be to start a new order and see if they are there.