The best methods would be to:
1. Improve your target customers (the people who are receiving your emails) - and not just mass mailing.
2. Personalize the email by using new technologies that adapt each email to the user - and not just by adding "Dear [name of user]. Here's a link to one of the leading service providers in the field: campaign-genius.com
(I happened to actually help them through Clarity, and know that they are really professional).
I've successfully helped over 300 entrepreneurs and would be happy to help you if you need. Just send me some background info by email before, so that the call can be 100% dedicated to giving you my advice.
Good luck

1. Yetmaill
2. Mailtrak
3. Watch your open-rates triple ;)

All the guys who answered before me had great points! Here's one I didn't see menitoned yet:

----> Send to your to your most engaged recipients first.

The more reliable engagement and open rates from this cohort builds positive reputation, for lack of a better term, with inboxes. It will increase the chances of inboxes receiving the next rounds of emails subsquent to those first successful deliveries.

I use and recommend ActiveCampaign to my clients. They have amazing automations, a nice CRM, and great deliverability. Schedule a call and I'll be happy to walk you through whatever you need.