New Session button unexpectedly cancels current session

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

When a session is running in the Claude Code (for vscode) sidebar and I start a new session, the existing session is cancelled / abandoned. The expected behaviour — consistent with how competing tools like Windsurf handle this — is that starting a new session simply opens a new session alongside the existing one, leaving it running.

<img width="560" height="1101" alt="Image" src="https://github.com/user-attachments/assets/d383dad6-5d0f-4c1f-af0e-25bdbbfe08fb" />

The workaround is to use "Open in New Tab" from the Command Palette before starting a second session, but this is non-obvious and adds friction to a common workflow.

Proposed Solution

When a new session is initiated from the sidebar while an existing session is active, preserve the running session and open the new one alongside it — either as a new tab or by adding it to the sessions list — rather than cancelling the current one.

This makes parallel sessions a natural part of the default workflow rather than something that requires foreknowledge of the tab system.

Even better, add support for opening a new session with a new Worktree also.

Use Case

Firing off a long-running agentic task and immediately starting a second task without waiting for the first to complete, without needing to reach for the Command Palette or the --worktree CLI flag.

Related Issues

• #7506 — Multiple Claude Code instances in the same workspace causes IDE integration to drop