You should try writing to the schools principal directly to enquire if they have vacancy. Sometimes, if they have the vacancy, they can allow students to join in at any part of the year.
To find out Principal's email address... you can try this method.
1) Go to www.sgdi.sg
2) On the left hand side, under organisation, type 'Ministry of Education', for Job title, type 'Principal' then click find
3) With the list of names generated, you can look for someone who is a Primary School Prinicpal... click on his name and a whole list of Principals from various schools will be reflected.