For new Subnets to have routing outside the VPC, ensure that they have appropriate inbound and outbound routes in a route table. Update the master route table (which is implicitly assigned to your new subnets), or define a new route table and explicitly associate it with the subnets. In this example, an explicit route table is created and the MGT, PKT0, PKT1 and HFE (if required) subnets are assigned to it. Create separate route tables for each of the MGT, PKT0, PKT1 and HFE subnets if desired.
AWS uses the most specific route in the route table that matches the traffic to determine how to route the traffic (longest prefix match). You need the rule to route all the non-Virtual Private Clouds (VPC) traffic to the internet gateway or ensure that the internet traffic is routed through your own NAT instance or Gateway.
For example, the following route table has a route for IPv4 Internet traffic 0.0.0.0/0 that points to an Internet gateway. The Local route covers any traffic destined for a target within the VPC (10.0.0.0/16) and therefore, is routed within the VPC. All other traffic from the subnet uses the internet gateway.