You can make a formula URL field button called ADD Similar which will simply do what the "Copy this Record" does using the More menu when viewing a record.
URLRoot() & "db/" & Dbid() & "?a=GenCopyRecord&rid=" & ToText([Record ID#])
You set each field's properties as to if it should be copied when using this feature.
You could have a report, perhaps even these users' Dashboard Report, for My Recent Records, which is filtered by where the record owner is the current user, most recently created records first. So they would look at a recent record, click the add similar button in the report and just fill in the missing fields were set not to carry over.
Here We Go Again
The demo has been created to allow you to add records.
The query would have to be modified slightly to grab the most recently created record by the current user but there is no point in putting that logic into a public demo as I have to allow everyone on the internet the ability to add records.
I entered Debbie Harry of Miami Florida as the first record as I was listening to Blondie. If you have any questions or need any help please feel free to call me using the information in my profile. Enjoy ...
Blondie - Call Me