I am not 100% what you mean by a splash screen but, based on what you have I believe you want a page that overlays your page once you click a button or link as you have done with the Header Widget, if that is not the case I imagine you want a pre-loader page so I will give you two replies and you can clarify what you need.
1- In case you need a page that overlays your with some information.
1a) You can use X Lightbox Functionality. You will create a new page, select on WP Classic backend editor, under the blue update button > page attributes > page templates > Select: Blank No Container No Header No Footer
You will than edit this page normally with Cornerstone and create your Splash page.
After that on your home page you will add a RAW Code Element with a Lightbox Selector, you can add this:
You will than add a Link, it can be an image that is linked, a button or a linked word, and add a classs to this link :
splash-me and add the page you have created to the HREF
That way when you click this link it will load the page you created as an overlay.
More info here: https://theme.co/apex/forum/t/shortcodes-lightbox/165
1b) Another alternative is, if you wish to do this but have a smaller box with some capture box for email for example, Convertplus is a good solution.
The plugin is available among our Addons. You can setup so it loads as soon as the client visit the page, or, you can set it to be launched from a button.
Please see here: https://www.youtube.com/watch?v=53evC1A-4UE&t=4s
2- If you are looking to have a pre-loader effect you would need to find a 3 party plugin.
Hope it helps