You could use the HotDocs Database Connection to connect with the TM database, with writeback enabled for the relevant HotDocs variables. This would function in a similar way to TM/HD Answer Source Integration, but the writeback could happen automatically (without user intervention). When the answer file is initially created, you could pass a unique identifier for the Contact record (behind the scenes) to HotDocs by linking a variable to the field. The Database component would then be filtered on that identifier so that the record in question would automatically be selected for subsequent assemblies.
One limitation of the writeback functionality in the Database Connection (which is also true of TM/HD Answer Source integration, I believe) is that HotDocs will not write back to an empty field. In other words, if the Time Matters field was empty to begin with, entering a value in HotDocs will not result in that value being written to TM.