Yes.
The aerial is a piece of wire that 'catches' the broadcast radio waves. In theory you can tune the length of the aerial to the broadcast freqency/wavelength.
Radio hams advise that the aerial should be nominally a half- or a quarter-wavelength.
DAB in the UK is around 225MHz, which corresponds to a wavelength of 1.33m/ Thus, the nominal length of a quarter-wavelength aerial would be about 33cm (just over a foot in old money).
That basically gives you the minimum length you need. IN broad terms, the longer the aerial, the more signal you are going to catch.
Bottom line, just add a wire to the aerial. Make sure there is a good electrical connection between the pull-up aerial The crocodile clip is an excellent idea. The wire that can be as long as you like, but if you want to be geeky about it, make it an even multiple of 33cm long (33cm / 66cm / 133cm / 200cm etc)
Good luck