I have a good recommendation, but you're going to have to take that with a grain of salt
I work for InMotion Hosting, and not only do we allow for lists being sent out from phpList to exceed 1,000 mails a day, but they've also allowed me to play with phpList in-depth on our service and I wrote up an entire phpList tutorial
series on how to use phpList. I've also done extensive testing to ensure it works well on our shared servers and up, and you can use the external SMTP service as well.
I used to be a System Administrator until I joined our Customer Community team, and now I write technical documentation with screenshots and steps to help explain problems for users of a wide variety of web hosting stuff that I've encountered over the years. I also get to reach out on forums like this to expand my own knowledge on topics and to look for other things to write about to help out our customers and the Internet in general.
We do have a default beginning 250 messages per hour limit on our shared servers, but with phpList's great rate limiting features you can easily stay within that 6,000/day (250 messages x 24 hours) limit. If you do have a much larger legitimate list that you need to send out to, you can also request a bulk mailing limit increase up to 750 messages / hour (18,000/day) on our shared servers.
Anyways hope that information was helpful to you, if you had any other questions about our hosting I'd be glad to answer those for you as well. If you were interested in signing up, let me know and I can hook you up with a coupon code as well to get a good deal.