A family of Microsoft word processing software products for creating web, email, and print documents.
In Word, each section contains three headers and three footers: a "primary" header and footer, a "first page"pair, and an "even page" pair. Which one appears on a particular page depends on the settings you choose in the Header & Footer ribbon. Read Using Headers and Footers for explanation.
In this case, you should check "Different First Page". Then copy the large header from the primary header (which will start to appear on the second page) and paste it into the first page header. On the second page, enter the smaller header that you want on that page and all the following pages. John may be correct about fixing the line spacing if the primary header is still too large. If your document has more than one section, you'll also need to know about the Link to Previous function.