When hitting governor limits due to the volume of operations on the Order object, moving automation to asynchronous Apex (e.g., using future methods, batch Apex, or Queueable Apex) is often necessary. This allows for more efficient handling of bulk operations and reduces the chance of exceeding governor limits. Salesforce provides guidelines on bulkifying code and using asynchronous operations to manage large data volumes effectively, which is essential for high-volume order processing.