Hmm, on http://www.workingintheuk.gov.uk/working_in_the_uk/EN/homepage/schemes_and_programmes/hsmp.html, it says:
If you have any general queries about the Highly Skilled Migrant Programme, please contact the Border and Immigration Agency Customer Contact Centre:
0114 207 4074
If you have a case specific enquiry, please contact the HSMP team:
0114 207 6020
So I thought the latter would be the correct number. I'll also try the other one tomorrow. Thanks.