The key is the VariableSetter and VariableRetriever transformers.  One way to achieve this:
  1. Set up your first source dataset.  The first dataset in the Navigator pane is the database dataset which reads from the table holding the start count value.
  2. Attach a VariableSetter to the source feature type from the database, setting the value of the variable to the attribute holding the start count.
  3. Set up your second dataset.  This dataset contains the features that need to be tagged with unique id's using the Counter.
  4. Attach a VariableRetriever to the source feature type from the second dataset.  This will receive the variable from the VariableSetter.  The "Attribute Receiving Value" now holds the Start Count.
  5. Attach a Counter to the VariableRetriever.
  
Note that for this to work the database must be the first dataset read by the workspace (be listed at the top of the Navigator pane).  This ensures the variable is set by the VariableSetter before it is read by the VariableRetriever.