Why the shipping methods you're expecting may not be showing up on the Cart or Checkout pages.
If you're implementing one of Eniture Technology's shipping quote products and it isn't displaying options on the WooCommerce Cart or Checkout pages, one troubleshooting step is to look at your WooCommerce Shipping Zones.
WooCommerce progresses through your shipping zones in the order they appear on the WooCommerce > Settings > Shipping page. WooCommerce stops progressing through the zones when a shipping rate is retrieved.
WooCommerce stops progressing through the zones when a shipping rate is retrieved.
In the illustration above, if the shopper's shipping address has a postal code that begins with "303", only the option for Free Shipping will be offered. The shipping methods for the zone "USA" won't be offered.
If the order of the zones is reordered so that the "USA" zone is listed first...
...then only the shipping methods for the "USA" zone will be offered even though the shopper's shipping address has a postal code that begins with "303".
If you want the shipping methods that appear in the "USA" shipping zone in the illustrations above also to be offered in addition to the offer of Free Shipping when the shopper's shipping address has a postal code that begins with "303", they must also appear in the "Local" shipping zone, and the "Local" shipping zone must be listed first.
Setting up Shipping Zones (Official instructions from WooCommerce)