Since some SMS are sent successfully but most of the SMS are resulting with the error:

Unknown error attempting to reach phone

I would strongly suggest that you contact AWS Premium Support to help you identify why SMS sent are not being delivered.

I had the same issue and I solved this way

Context

Allow AWS Cognito send SMS messages for SMS MFA codes.

Solution

Step 1

1. Open the AWS Support console
2. Create a new Service Limit Increase case

Here's an example possible values

• Limit Type -> SNS Text Messaging

• Fill in all of the optional fields, even if the system's url is your test environment.

• Use case description -> AWS Cognito is indicating me that "To send SMS messages for verification or MFA, you must request a spending limit increase".

• How do your customers opt to receive messages from you? Provide specific information about the opt-in process -> When setting up an account a user is asked to setup their Multi-factor authentication.

• Please provide the message template that you plan to use to send messages to your customers -> "Your authentication code is {####}"

• Region -> Select your desired region

• Resource Type -> General Limits

• Limit name -> Exit SMS Sandbox

• New limit value -> 1000 (that means 1000 USD/month)

AWS Documentation for this step

Step 2

1. Open the SNS console
2. Open Mobile > Text Messaging (SMS)
3. Click Edit, on the Text messaging preferences panel
4. Set the Account spend limit, it must be any value betwee 1 and the limit you requested to AWS Support, in my case $1000.
5. In same view, scroll down and configure the Delivery status logging, create or select and existing Role.

AWS Documentation for this step

Conclusion

You need an AWS limit increase and also configure SNS correctly.