Can I restore my SQL Server database to a point before the last full backup with my current backup strategy?
Yes, you could do a point-in-time restore.
You would need the last full backup and all log backups up to the point you need, as log backups do not contain all earlier logs (they are in one sense incremental, rather than differential).
For a week's worth of logs on a busy database, it may take some time to restore. Therefore a better solution (if you want to stick to weekly fulls) is to do a differential backup every day or twice a day.
This means that you only need the last full backup, the last diff backup, and all logs going forward up to whenever you want to restore.

A fairly standard backup cycle on many servers is: full daily, diff 4-6 per day and logs every few minutes. This gives a pretty fast restore time. It also means more importantly, that your maximum data loss is never more than a few minutes, rather than up to an hour.