Unfortunately it's unlikely that that there's an answer to this; according to the prologue, the hobbits themselves didn't keep records until settling the Shire (T.A. 1601 according to Appendix B), and weren't much for learning and history in any case:
A love of learning (other than genealogical lore) was far from general among them, but there remained still a few in the older families who studied their own books, and even gathered reports of old times and distant lands from Elves, Dwarves, and Men. Their own records began only after the settlement of the Shire, and their most ancient legends hardly looked further back than their Wandering Days.
Fellowship of the Ring Prologue Chapter 1: "Concerning Hobbits"
This is quite a long time before Sméagol's birth (he takes possession of the Ring in T.A. 2463), but his people were isolated from the Shire-folk; Gandalf notes in Fellowship of the Ring that Sméagol's people had settled on the Anduin, near the Gladden Fields:
'Long after, but still very long ago, there lived by the banks of the Great River on the edge of Wilderland a clever-handed and quiet-footed little people. I guess they were of hobbit-kind; akin to the fathers of the fathers of the Stoors, for they loved the River, and often swam in it, or made little boats of reeds.
Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"
Based on Appendix B, this group probably split off from the other Stoors in about T.A. 1356:
1356 About this time the Stoors leave the Angle, and some return to Wilderland.
Return of the King Appendix B: "The Tale of Years" The Third Age
All of this is a roundabout way of saying that there isn't an abundance of information regarding contemporary names for hobbit-folk. Considering their long isolation, it seems likely that they developed a unique name for themselves, which was lost when they died out.
The most likely origin of this name is probably the Sindarin Periannath; the first reference to that word being used is in T.A. 1050:
1050 The Periannath are first mentioned in records, with the coming of the Harfoots to Eriador.
Return of the King Appendix B: "The Tale of Years" The Third Age
But then again early hobbits likely had a name for themselves that pre-dated their first encounters with the Big Folk. The prologue kind of suggests that the tribe-names (Harfoot, Fallohide, and Stoor) were in use at the time, but it's by no means clear on the point:
Before the crossing of the mountains the Hobbits had already become divided into three somewhat different breeds: Harfoots, Stoors, and Fallohides. The Harfoots were browner of skin, smaller, and shorter, and they were beardless and bootless; their hands and feet were neat and nimble; and they preferred highlands and hillsides. The Stoors were broader, heavier in build; their feet and hands were larger, and they preferred flat lands and riversides. The Fallohides were fairer of skin and also of hair, and they were taller and slimmer than the others; they were lovers of trees and of woodlands.
Fellowship of the Ring Prologue Chapter 1: "Concerning Hobbits"
Unfortunately, we just don't know.
