Snowflake/ASW S3: IAM and AWS Policy STS conditions

Question

I have set up an S3 bucket for csv file loads for ingestion via a Snowflake stage. I I completed all the steps to achieve Snowflake Storage Integration to Access Amazon S3 and it all works fine.

I now wish to set up the same Storage Integration for another Snowflake account to the same S3 bucket. I noted that in the policy document setup in AWS, it contains the Principal value for the snowflake_user_arn and the condition sts:ExternalId for the snowflake_external_id. Can I alter that Policy to include the additional Snowflake Integration references, or should I simply create another Policy Document? If the latter, can I use the same AWS Role I set up for the first Policy on this bucket?

Sorry if I have failed to provide not enough or too much info in the question, I wished to cover all bases.