The official Jira Cloud for Slack connector has a couple configuration options. In the individual subscriptions, you can toggle between the longer "Show entire text for summary, description, comment, and worklog":
and "Truncated" message style:
These options are configurable on the subscription page, which is easy to get to by typing /jira manage in your Slack channel, and then clicking the Manage button for the relevant subscription.
If you require more granular notification types, some of the other Slack connectors on the Atlassian Marketplace provide options to change what fields appear in the message through a web interface or with a JSON template.
Daniel | Atlassian Support
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