1- On what basis you are applying ILR ? Which form you filled in i.e SET (M) or SET (o) ?
2- if you have submitted your online application within 28 days of completing your qualifying period than you would be fine Even your PSC appointment is after expiry of your existing LTR because in case of online application, the Date of application is considered when you submit your online application but not the PSC appointment day, I think you should be fine .
3- Other option you have to try hard to keep checking and looking for earlier PSC appointment if someone might cancel their appointment and you can reschedule your appointment but this would be extremely difficult , you would have to be very flexible in regard to locations in whole Uk and dates ?
4- Other option is you can Vary your ILR application from PSC to postal application, send them your online completed printed application form along with supporting documents by recorded post , including covering letter explaining whole situation that you want to Vary your PSC appointment into postal application. wish you good luck

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