Invisalign tray is only half way over my teeth. It doesn't fully cover my teeth. Will it still keep my teeth straight?

There are times when new impressions or a scan is indicated to remanifacture aligners  to avoid the areas where sores or discomfort was felt.  This will many times lead to better tracking and quicker treatment compared to trying to adjust and trim the aligner in office.

as long as your teeth fit into the aligner, the way a finger fits into a glove it is ok. if not please see your doctor

If the aligner/tray does not fit precisely on the teeth they may not move as prescribed. Often times this is referred to as 'not tracking ' and the trays may need to refit for the most desirable progress.