IMHO your best bet is really to avoid using mailchimp for opted-in lists - at least 50% of the email I get from them is spam (the messages are also generally illegal under UK law), so I would wager that a large number of services proactively dump all mailchimp email into spam folders. A lot of other people seem to feel the same way, here's a few examples:
https://lists.debian.org/debian-isp/2016/08/msg00001.html http://www.techie7.com/threads/72825/ http://otakunozoku.com/block-that-mailchimp-spam/ http://www.warriorforum.com/main-internet-marketing-discussion-forum/614184-mailchimp-haven-spammers-discuss.html http://bitmonger.blogspot.co.uk/2016/02/avoid-wretched-spammers-at-mailchimp.html http://forum.spamcop.net/topic/16829-internetwide-block-for-spamchimp/
There is one service that I actually want emails from who use mail chimp - if not for that, I'd be auto-marking all mailchimp emails as spam.
If, as you say, you're running on entirely opt-in lists then your best bet may be a self hosted solution. Depending on your volumes, this may be more expensive than mailchimp.
Alternatively, according to a survey done last year by Capterra, MailChimp alternative Constant Contact had much greater deliverability than MailChimp:
Of our survey participants, the primary reason that marketers preferred Constant Contact when they had used both systems was because they saw greater email deliverability with Constant Contact.
Anecdotally I don't believe I've ever received a single spam from Constant Contact, so that gels with the survey results.