I lost my packet of birth control with 4 active pills left. Should I wait until my next pack should start or start the next pack already?

Should wait... Missing only 1 pill is not likely to change thing much and so I would just wait to start your new pack as usual. Maybe, just maybe your period will come 1 day early. Otherwise, it should not change much at all. If you have any other trouble or additional concern, consult doc. Good luck.
Make up the 4 pills. With the first 4 pills from the new pack assuming u are taking a monophasic pill, if u are not late, meaning u have not missed any pill days and call your obgyn as if u were my pt, I would give u another sample pack, after your period then start a brand new pack, but u should call yoor obgyn and ask even on the weekend.