As per your statement, i understood that you are looking for an manual Interaction Switch from a button configured on the composite.
We can use Invoke Action API to switch the thread manually. This Invoke Action is available in the Action Tab of any Button Configuration. We should pass the Interaction .pyExecute Page as a parameter to it. This .pyExecute page would be available in D_CPMPortalRecents.
Can you provide more details on bleow of your requirement.
"make the link availble in all opened task page too so that agent can click and open the right customer from any screen" - Does it mean you would like search again for any custormer from any service cases context (means - when you are processing a case in interaction or directly from 360 degreee view of interaction)? If So, I've two questions to you :
1. You would like to search for another cusotmer in same interaction context - means you will loose current customer context
2. Or, you would like start another interaction right from 360 degree view
I don't think you require to swtich the thread until there is no OOTB way. I would suggest not to play around swtiching threads manually.