Postgres WAL logs-- point in time recovery with (time) gaps in the logs?
PostgreSQL does not care when the WAL logs are generated, it just reads them sequentially on restore/recovery.

If the WAL logs were properly sent after the restart of the database, they should be restored as well.

However, if there is even a single WAL file missing for some reason (in this example, I'd guess that'd be either the last segment before the shutdown or one or more of the first ones from after the startup), which is typically either an incorrectly configured archive_command (missing error checks for example) or simply a broken archive server, the restore process cannot go past that point.