There is not a configuration option to have screenpops fire on the connected event, and we have no plans to add such a configuration at this time. That said, with the newly released client extension points and the lightning messaging service, the tools exist to attempt your own customization that intercepts the screenpop method, listens for the conversation connected event, then fires the new screenpop event.
With all that said, this sounds more like a status management issue, where the Agents are in a status that would allow them to receive conversations from the ACD system despite not actually being ready to accept conversations. It would likely be more effective for the agent to be in a non-acd status that reflects that they are doing additional work (either remaining in ACW, or using a status like "Busy- case work", either of which would prevent conversations from being routed). The lightning messaging service events would be an excellent way to automate this, using the status events to automate their Genesys Cloud status based on the work they are doing in Salesforce; if they are on a record that requires additional work time, their status in Genesys Cloud could be automatically set to "Busy-case work" until they complete their work and close the Salesforce object, which would then trigger an event to set their Genesys Cloud status to "on queue".
Relevant links:
https://help.mypurecloud.com/articles/events-in-salesforce/
https://help.mypurecloud.com/articles/extension-points-in-genesys-cloud-for-salesforce/
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Richard Schott
Genesys - Employees
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