You did not specify the exact ticket type that you have, but it sounds like you have the "DSB Standard Ticket". In that case, the answer is yes, you're allowed to explore the city and take another train.
DSB description of the standard ticket mentions:
Billetten kan bruges til en hvilken som helst afgang på den pågældende dag til den valgte strækning
It means that the ticket is valid for any departure on the ticket's validity date, in the chosen direction.
There's some additional information further, though:
Når du rejser med en Standardbillet, kan du selv vælge, hvilken afgang på dagen der passer dig bedst. Vil du være sikker på en siddeplads, så køb en Pladsbillet til en bestemt afgang. Særligt omkring ferie og højtider er der mange passagerer i toget, og du kan derfor kun være sikker på at komme til at sidde ned, når du har Pladsbillet.
It says that, to reserve a seat, you buy a Pladsbillet (Ticket with place reservation), and those are for a specific departure, which makes sense - you'd get a seat reserved on one train, not all trains of that day. This should not be necessary for most of the year.
So unless you see a specific seat on your ticket (and a specific departure time), you can choose your own departure time.
As an additional note, should you have any questions during your journey, all rail company employees at stations or onboard the trains speak English.