Yes — contraception is well suited to remote consultation, and desogestrel is one of the more straightforward prescriptions.
What the consultation covers: blood pressure and weight, migraine history, smoking, family history of clots, current medicines, and whether there is any chance of existing pregnancy.
Desogestrel is available from some pharmacies without prescription, which may be quicker for a straightforward continuation.
What we will always mention is that the pill is not the most effective contraception available. Implants, coils and injections are more effective precisely because they do not depend on remembering anything daily. If you want the most reliable option, we will say so rather than simply supply what was asked for.