I am not sure how you are currently working (architecturally) however can i suggest you explore injecting the time using a system prompt when the call is connected. This is the way I have been injecting data that is relevant to the call that gets picked up and seems to work without fault, for example if you were using make.com you can receive a status on your server url of "in-progress" and then using and HTTP post request inject the time (or other data) using the control URL - with a payload like this:
{
"type": "add-message",
"message": {
"role": "system",
"content": "the current date and time is:{{133.
date/time
}} "
},
"triggerResponseEnabled": true
}