It would be possible to restructure the Supreme Court because article 3 of the US Constitution is very vague on how the Supreme Court should be structured in the first place. Pretty much all it says is that there should be a Supreme Court without providing much further detail. Therefore, it is possible for Congress along with the President to make changes to the Court through legislation.
Indeed, this has happened several times already. The original Supreme Court had six justices. That number grew to ten during the early and mid-nineteenth century. Its current nine-member configuration came about under the presidency of Grant in the 1870s.
Whether or not it's a good idea to restructure the Supreme Court is a matter of opinion. Some people feel that the Supreme Court has become too partisan in recent years. Many feel that this has eroded some of the democratic nature of the country since the judges are not beholden to the electorate. Adding more justices could potentially restore a nonpartisan balance.
There have also been proposals to limit the tenure of the justices. It is unclear how constitutional this change would be, as article 3 dictates that justices "shall hold their offices during good behavior." This has usually been interpreted to mean that they have a lifetime appointment. Ironically, it would be up to the Supreme Court itself to decide the constitutionality of term limits if such legislation were passed.
On the other hand, there are arguments against restructuring the Supreme Court. Passing legislation to change the number of justices would likely benefit one political party over another. Just the act of making these changes could be considered partisan and erode faith in the justice system.