This 'might' be possible, but would require quite a bit of effort (and may not ultimately work or be supported).  Out-of-the-box, the Show Other Crews/Jobs command ultimately results in the broadcastunitsevents.fmt returning minimal event/crew (and XY) info in an internal 'Other Jobs' XML file that TC maintains.  Then, once the Other Jobs are returned, the Get Other Job info allows a map click on the otherJobs Pin type, and then gets the Other Job info (returned from the broadcastunitsevents.fmt) for the selected otherJob Pin from the internal Other Jobs XML file and displays it in the smaller form in the bottom of the map.

To attempt to do similar for jobs already assigned to your crew, you would have to mimic the code in the ..\default\jobinfo.htm file, but instead try to use selectPinRequest for 'assigned' jobs (from mobileInserviceBitmapList.xml), which you would likely need to add to mobileInserviceSettings.xml in the <pin> section (similar to <otherJobs>), and then, if the assigned Pin can be clicked, and if the assigned Pin contains the job ID, cycle through the internal Jobs XML file that contains all the Job data that came down via dispatch.fmt when the job was initially assigned.

The key here would be searching through the internal Jobs XML vs. the internal Other Jobs XML.  Theoretically all the data is present with regard to the Job data, and there are definitely map Pins for the dispatch assigned jobs.  I'm just not sure all the information needed is present in the assigned Pins, and/or if the selectPinRequest supports map clicks for assigned Pins.  As a brainstorming exercise, this approach that seems plausible if all things align.