It's quite common for previous searches to appear in a drop down list below the search boxes as soon as focus goes into the box - or even for a dedicated search page, simply to automatically show the saved or recent searches in a list without having to manually make that happen.
This then allows the previous searches to be sensibly filtered or ordered as you continue typing.
In such a scenario, the search box needs to be as high as possible in order for the drop down not to appear (and be instantly hidden) underneath the keyboard.
It's also common once the list appears, for an item to allow access to manage the saved searches - eg to clear the list.
You'll see this on a variety of applications and browsers on Mac / Windows - and on mobile.
If not showing the list automatically, then to aid discoverability you could move the icon inside the search box so that it's clearly related to the search box rather than a separate chunk of functionality. Having said that, there's usually a 'clear' button inside the right end of the search box, which you haven't included in your mock up...
In any case, although not so important to the main question, I wouldn't use the diskette symbol, perhaps a clock symbol might work better, and then display the age of the saved searches in the list. It's possible to combine 'best matches' and 'saved searches' in the list with separate symbols like in the example below.
Note the clever combined symbol of the magnifying glass and the clock in the first image.
Yeah - grabbed these off Google Images - sorry about the weiner item :-)