Tb h I tinkthey seriously could have taken Egypt and Hungary. With a bit of luck they might have used the former to pull an Arab conquer and get all the way to Iberia, which is at this time shattered into several tiafas.
The Mongols didn't conquer Mesopotamia till 1258, with the Siege of Baghdad. So it isn't till after that that they could focus on Syria and Egypt, however by that point the conflict between the Golden Horde and the Ilkhanate was already building up. So while I'd say a few more fortunate encounters could allow the Mongols to avoid the Battle of Ain Jalut, conquering Egypt would be a stretch since Baybars and other prominent Mamluks were already active in the Egyptian military against the mongols.
However I would say the mongols could have had additional success in Europe during the 1240's if Ogedei hadn't died. No one was coming to Hungary's aid, I think the Duke of Austria actually attacked after the Hungarian army was destroyed to claim a disputed county, and Poland too was divided. An important aspect for the Mongols would be gaining local allies, as they were already overstreched, and in this case Emperor Frederick II and the Popes were already feuding. While neither side was stupid, the Mongols were very good at taking advantage of local issues. Swear friendship with one side, promise to treat Christians well in their territory, 'give' Hungary back under the control of King Bela (in reality turning Bela into a Mongol vassal), and then help one side decimate the other. The Mongols brutally sack a number of cities, ensuring that side stays down, finds some insult dealt them by their 'ally' and then use their superior mobility to ravage the land of their ally and even destroy their army by surprise. Doubt it would go further than Hungary and/or Poland being vassalized and Germany and/or Italy being raided and even losing part of their eastern lands as Batu Khan was feuding with Guyuk who succeeded Ogedei. After those two the other khans wanted to direct their attention to richer land in southern China and the Middle East, so any further gains would become impossible and their vassal kingdoms in Europe would probably break free.