You could use Ajax to call Rest API to filter the announcement items.
/_api/web/lists/getbytitle('anno')/items?$filter=Created gt datetime'2020-09-01T00:00:00'
This Rest API will return all items created after 2020.09.01 00:00,you need to replace it with a js dynamically calculated value.Then you can show the latest item value in page if this Rest API returns values.
If your main page is a classic page,you could add a script editor in your page add script in it directly.
If your page is a modern page,you need to create a SPFX to achieve this.SPFX Document
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