I will have a standard "request" form that gets displayed to the user when they make a new request.
Is there a way to make a second form attach to this same list item so that administrators could see more information than the basic request would show?
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I think you can only have on form, but I would suggest another solution for your requirement. With the help of the "Expression" choice in the appearance section of the controls and the inline function "fn-IsMemberOfGroup" you can configure the controls in the way, that there's a check if the current user is a member of a specific permission group or not. If he's a member he can see the control otherwise it's not visible.
So you could put all your administrators in a specific SharePoint permission group and the configure the desired controls like in the following screenshot.
Good solution, and to extend on this, you could apply the fn to a panel containing your controls so you don't need to apply this to every control, just group the controls to be hidden in a panel and they'll appear when an administrator opens the form.
You can only have 1 form per list item though. Tabs and panels are a nice way of giving the impression of a multi-stage form, which it sounds like you are after.