In order to get tickets for the extra performances, theatregoers are being asked to enter a ballot. Winners will then be able to book up to two tickets for their selected performance.
The deadline for entries is midnight on 19 January.
Hiddleston, whose previous stage credits include Ivanov opposite Kenneth Branagh (Donmar West End), is joined in the cast by Deborah Findlay, Hadley Fraser, Mark Gatiss and Birgitte Hjort Sørensen. Donmar artistic director Josie Rourke directs.
There is also a draw to win a weekend stay at the Radisson Blu Edwardian Mercer Street Hotel, along with tickets to the production.
We've teamed up with our WhatsOnStage Awards Official Hotel Partner, Radisson Blu Edwardian to offer a luxury hotel package for the 14th annual WhatsOnStage Awards, which takes place on Sunday 23 February 2014 at the West End's Prince of Wales Theatre.
The package includes your overnight stay in the Radisson Blu Edwardian Mercer Street Hotel or Radisson Blu Edwardian Hampshire Hotel, both including full English Breakfast, premium ticket to the Awards Concert (normally £90 each) plus a souvenir programme.
The package is £195 per person based on two people sharing a double room. Places are limited so book now!