Thank-you for your question. Tier 4 student applications can be made either during a period of existing leave or after a period of leave has expired. This follows from the requirement in rule 245ZX(b) that "the applicant must have, or have last been granted, entry clearance, leave to enter or leave to remain [in one of various categories of which student is one]". Therefore you are entitled to submit a new application. There is no time limit for you to do so. However, one of the substantive requirements for leave to remain as a Tier 4 student is that "The applicant must not be applying for leave to remain for the purpose of studies which would commence more than one month after the applicant's current entry clearance or leave to remain expires" (rule 245ZX(l)). Therefore in order for your application to succeed, your course of study must not commence more than one month after the expiry of your current leave to remain in the UK.
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