It sounds like you're looking to implement a specific on-call escalation path for your assignment group, where if the first person on-call is unreachable, the next person on the list should be contacted instead of continuing to rotate through the first group.

One way to achieve this is to set up an escalation policy within your incident management system or on-call scheduling tool. You can define the order of escalation, specifying who should be contacted first, second, third, etc. You can also set up rules for how long to wait before escalating, or how many attempts to make before moving to the next person on the list.

In your case, you could set up an escalation policy where the first person to be contacted is the person on the Gold shift, and if they're unreachable, the next person to be contacted is the person on the Red shift. This way, if Engineer-X is unreachable, the next person on the list to be contacted will be Engineer-1, regardless of which shift they're on.

There are many incident management and on-call scheduling tools available that allow you to set up and customize escalation policies, such as PagerDuty, OpsGenie, and VictorOps. You may want to explore these options to find a tool that meets your specific needs.