The evacuation was very well organized. The primary route was through Yugoslavia and, surprisingly, Italy. While Yugoslavia was still neutral in 1939-1940 and transit was straightforward, Italy was already an ally of Germany, even though they did not participate actively in the war effort yet. Italian people, including administration & diplomatic staff, were sympathetic towards Poland. Italian minister of foreign affairs Geleazzo Ciano was a good friend of Polish ambassador, and he agreed to treat transit of Polish soldiers as strictly civil matter of "workers transfer". Italy even provided direct train Postumia (Postojna)-Milan-Modane, and Polish "workers" were able to buy tickets on credit. This allowed to evacuate 73-75% of soldiers from Romania and 52% from Hungary.

As per memories of Franciszek Skibiński, soldier of the Polish 1st Armored Division:

"We decided to escape [...] to Yugoslavia and then to France. [...]
On the evening of 23 September we were already in train going towards Yugoslavian border. [...] Transit through Hungary and Italy - not to mention Yugoslavia - was surprisingly easy. We weren't searched. We started regretting abandoning our uniforms left in Budapest. [...] We ate breakfast in Trieste, dinner in Genoa [...] Next morning we were in Ventimiglia, in France"

A very good summary of the evacuation, including secondary route to French Syria, is available here: https://warhist.pl/polska/ewakuacja-zolnierzy-polskich-z-rumunii-i-wegier-do-francji (in Polish).

Sources:
Franciszek Skibiński "Pierwsza Pancerna".