If you set the 'Report Generation Speed' to 'Generate reports immediately, reports can be overwritten until due date' then students can continue to submit assignments until the due date overwriting previous submissions - originality reports are available after 24 hours for second and subsequent submissions.
If you require an audit trail of submitted assignments then as suggested in other posts you need to create more parts, please bear in mind that TII treat multipart assignments as just that an assignment with up to 5 parts not an assignment with 5 submissions of the same piece of work.
When using 'Grademark' TII will apply a proportion of the marks to each part so a 5 part assignment would give a maximum of 20% to each submission.