Yes, it sounds like an independent trigger campaign would be the best first step. You could, though, have that be the Engagement controller instead of a qualifier; meaning, have the "welcome email 1" that gets triggered by SFDC campaign member, sends the first email, and then adds the person to the Engagement program in the appropriate stream. That way, if you really want to wait 30 days before the next engagement email, you can keep them outside the engagement program and then add them when their time qualifies—rather than using paused statuses.
One thing to keep as a note, if you have an email outside your engagement program streams, you'll need to merge reports or take that into consideration when calculating your overall Engagement Program successes.
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