First, let us not use the term "dpi". IT means dots per inch, not scanning. I shall use the term PPI because we are scanning pixels in this context.
The resolution of your prints is probably equivalent to about 200 PPI, and you should get very satisfactory print quality from files containing 240PPI at output print size. Therefore it may not make much sense to scan at greater than 240 OUTPUT PPI.
There is a distinction between INPUT PPI and OUTPUT PPI. OUTPUT PPI is the PPI per inch at the output size of the image. INPUT PPI is the PPI the scanner would create to get that value. If the Scale is the same (100%) between the Original and the Output (i.e. both are the same dimensions), then the INPUT PPI and the OUTPUT PPI are the same. If the Output dimensions will be larger than the input (Original) dimensions, then the INPUT PPI will have to be greater than the OUTPUT PPI if you want to have enough pixels to cover the larger Output dimensions at your required OUTPUT PPI.
You should also decide whether you wish to scan at 8 or 16 bit depth (which for the three channels is 24 or 48 bit depth). I recommend 16/48 bit depth. You'll get better quality files with smoother tonal gradations for printing.
At this bit depth, storage and file size considerations could make a difference to you as between scanning at 360 PPI or 240 PPI. At 240 PPI a scan which has an Output size of say 7 * 10.5 inches would make a 24.2 MB file, whereas at 360 PPI the same file would be 54.5 MB - this kind of difference adds-up over many files. Granted storage is cheap these days, but nonetheless something to consider.
Now, for your 2*3 inch Original, let us say you wish it to be printable at 240 PPI and at 7*10.5 inches (this maintains the original aspect ratio). Hence your output PPI would be 240. There is a scale factor of 350% between the dimensions of the original and the dimensions of the Output, hence to have your 240 PPI Output PPI, your INPUT PPI would be 900. If the software you are using allows you simply to state the OUTPUT PPI you want, then you need not concern yourself with this distinction, but good to know what's happening under the hood. If your software only allows you to specify INPUT PPI, then you need to take into account both the scale factor and the output PPI in order to know the correct INPUT PPI.