Yes.  You will need to be sure that you actually operate “exclusively” for charitable purposes and that your governing instruments include the required charitable provisions, including the requirement to use remaining net assets for charitable purposes upon dissolution if such obligation is not included in your state law.  You may have to amend your documents to fully comply.  You will then need to file a Form 1023 Application for Recognition of Exemption to request the charitable status.  Assuming you are successful, the exemption will be good as of the date of the application.

You can’t readily switch back to (c)(4) again, however, without leaving all of the assets in a charitable entity.  The switch is a one-way ratchet.