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

I have been managing CRM and Customer Acquisition campaigns for global brands such as adidas or Vistaprint so I am quite familiar with this topic.

In order to increase your open rates you can use several tactics:

- Continuously test your Subject Lines. You need always to keep testing since there is no secret recipe on creating the magic subject line
- Have a consistent brand domain/sender domain
- Prepare beforehand an email marketing plan with sending out dates/times, topics... it is crucial to have a valid strategy in place
- Study your consumer behavior and journey: open rate may be affected by that. For instance a consumer may be not willing to open a commercial email if he/she is still considering/evaluating your product
- Study the best time of day / day of the week where to connect with your consumers via email

Regarding getting a better in box placement, here you have few considerations:
- If you email in about a promotion, it makes sense that it should be within the Promotion tab. So review what is the promotional strategy of your competitors before planning yours
- Avoid using any concepts (text, content, domain, images) that may sound repetitive/spammy
- Plan a different inbox placement strategy for sender. Gmail should be your main focus but do not forget other providers (Hotmail. etc..)
- A review of your html email template needs to be done periodically to be always up to speed with new email providers functionalities. This will allow you to reach easily a better inbox placement.

I am happy to discuss all these points further and to give you useful tips on how to improve your CRM strategy and how to connect it with the rest of your digital strategy.

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.