Others will be better placed than me.
Do you have or plan to have and use any of the following though have you seen or contributed to your publisher's marketing plan? You might need to coordinate some of the following with your publisher.
Have you arranged/tried to arrange bookshop and school visits?
Do you have a website, blog, social media that you update with content regularly?
Do you have an email list?
Do you plan on any advertising? I'm told by self-publishers (I know you've a publisher) that Facebook ads can be productive.
Have you set up Amazon author pages, Wikipedia, Goodreads and so on. Picture books tend to be purchased in paper rather than kindle formats but Amazon still sells a scarily high percentage of paper books in the UK and USA.
Do you post on other blogs with posts that are more subtle than 'I've just published this' (I'm not saying you would be gauche but I see a lot of self-published authors who do post like this.
Where do your audience live? Could you advertise at local groups where parents and 3-6 year olds congregate?
Do you have a press pack? Have you contacted local press with whom you could do a meet the author/similar story?
Are there any marketing channels/events/groups that could tie in with the subject of this book or that you could hook onto press articles.
I'm sure you have but google some articles by self-published and other successful emerging authors in your book's age group for ideas.
Is your illustrator doing anything or could you coordinate with them?
Although it's a crowded field out there, if you are able to do some of the above (and I'm sure other suggestions will come along from other posters better placed than me) in coordination with your publisher's marketing department then it may help raise your profile.
Could the book have any international appeal/has your publisher launched in other markets? Different markets often use different social media for example.
Good luck.
Keep track of any expenditure for your accounts.