There is a theory that correlates the highest rates of emigration came from countries in Europe where the reduction in the birthrate was postponed the longest. France underwent the demographic transition before any other country in the world, a century before most of its European counterparts.
Below is the crude birth rates of France and the year at which Italy reached the number. It clearly shows that Italy's demographic decline was much later and much quicker when it did begin.
France
1801 32.9 (Italy 1913)
1815 32.5 (Italy 1920)
1830 29.9 (Italy 1924)
1848 26.5 (Italy 1931)
1870 25.9 (Italy 1931)
1890 21.8 (Italy 1949)
1900 21.3 (Italy 1949)
1913 18.8 (Italy 1965)
Below is a comparison of the crude birth rates of various European countries in 1913. It is no surprise that Russia was a leading country for emigration, along with Austria-Hungary, Italy, Spain and Portugal. In contrast, emigration from Germany, the Netherlands and Scandinavia had peaked during the 1880s. One thing to note is Ireland's low crude birth rate, this was mainly due to emigration from this country including a large number of young women (in contrast to Italy's being being over 75% male), the loss of a large number of women of the child-bearing age decreased the crude birth rate. But even Irish emigration had declined since the 1890s.
1913
Russia 43.9
Bulgaria 42.0
Romania 39.2
Serbia 38.5
Hungary 35.7
Italy 33.3
Spain 33.1
Austria 32.6
Portugal 32.3
Germany 29.8
Netherlands 28.6
Denmark 27.5
Norway 26.1
Sweden 24.7
Great Britain 25.0
Switzerland 25.0
Belgium 23.6
Ireland 23.3
France 19.6
Also, France was unique in that it possessed a largely fertile soil with an inheritance system that gave all children equal portions of land. This was enshrined by law in 1790 and again 1793. This was written into the Code Napoleon in 1804 and became widely accepted. By contrast, most other European societies possessed male primogeniture with eldest sons inheriting the majority or all of the land, and leaving nothing for girls. This gave the French an incentive to have fewer children to keep landholdings large enough for their children to survive. Another effect this had was that by 1913 France had a far larger proportion of its citizens as farmers than Britain, Germany or the USA (it still does today).
It was during the time of the Third Republic that mass migration from the rest of Europe to the Americas and Oceania was at its peak. France by contrast was importing large numbers of foreigners including Italians, Poles, Spaniards, and Eastern European Jews. In French North Africa it relied on Spaniards, Italians, Maltese, Swiss, and Germans to be the bulk of the colonists. Also, France's Republican model of citizenship and equality led to many of these immigrants who remained permanently assimilating into the French culture.
Also, by that time France had universal male suffrage, unlike much of Western Europe. In a society dominated by small property owners, the protection of these peasant farmers, along with small craftsmen and small factories, became the policy of most political parties. Beginning in the 1880s, the French government adopted tariffs to protect farmers including the banning of American meats in 1881. These culminated in the Méline Tariffs of 1892. French tariffs continued to protect French farmers and were largely supported by the broad spectrum of parties including Republicans, Royalists, Clerical and anti-Clerical parties.
In effect, France chose an economic model that would encourage protection of small farms and industries. Great Britain on the other hand embraced free trade since the abolition of the Corn Laws in 1846. Small farmers in Britain were effectively forced to compete with imported grain and foodstuffs, leading many to abandon farming all together, with many choosing to emigrate early on. This protected French peasants from abrupt change, but it also made many subsistence farmers, not in danger of starving, but with little disposable income. In Germany too, Chancellor Caprivi had embraced free trade, and this promoted German industrial growth to the detriment of agriculture. However, the consolidation of German farms into larger land holdings made it so that agricultural output was 25-50% more than in France despite poorer soils.
So to get France to have more emigrants, you'll need to maintain a higher birthrate longer.