During the whole series and visual novel I don't think it is ever mentionned if you need network on this phone to send a message or not. In the story the phone attached to the microwave is used as a relay to send a D-Mail, so it needs network to receive messages. However what would happen if the d-mail was sent directly from the original phone without network? We know that SERN is able to intercept D-mails thanks to their network monitoring but it doesn't mean that D-mails originally come from the mobile phone network. It's just there when it arrives in the past.
I think the D-mail actually alters the timeline into the one that has received the message. It doesn't send at that present time but it is sent in the past.
The sending of the message actually becomes the change between the timelines.
But this makes me wonder how the pager thing worked? Different timelines, different technologies. I mean what was the sender number in that message received by the pager? Is that really possible?
Please correct me if I'm wrong.
EDIT : (after OP comment) The answer is NO,the D-mail does not require network to be sent.