A wand is just another way of channeling magic, and that magic is channeled using the core which is encased inside wood, in case of a wand!
It's highly possible that Hagrid's wand's core is in the umbrella, but I really think the core would have been broken. I believe Dumbledore just went on ahead and did a
reparo like Harry did, in Deathly Hallows.
In Sorcerer's Stone, Hagrid takes Harry to Olivander to get his wand, but Olivander recognizes Hagrid (Not hard to do) and asks him about his wand.
"Rubeus! Rubeus Hagrid! How nice to see you again.... Oak, sixteen inches, rather bendy, wasn't it." "It was, sir, yes," said Hagrid.
"Good wand, that one. But I suppose they snapped it in half when you got expelled." said Mr. Ollivander, suddenly stern.
"Er -- yes, they did, yes," said Hagrid, shuffling his feet. "I've still got the pieces, though," he added brightly.
"But you don't use them." said Mr. Ollivander sharply.
"Oh, no, sir." said Hagrid quickly. Harry noticed he gripped his pink umbrella very tightly as he spoke.
"Hmmm," said Mr. Ollivander, giving Hagrid a piercing look.
Hagrid lied twice, his wand wasn't in pieces (not anymore) and he still uses it, but only under Dumbledore's permission.
Harry Potter wikia does suggest the possibility that Dumbledore might have repaired Hagrid's wand!
Following his expulsion, the Ministry of Magic forbade Hagrid from practising magic and destroyed his wand. Albus Dumbledore nonetheless convinced then Headmaster, Armando Dippet, to give Hagrid the job of Gamekeeper and allow him to remain a resident of the school. Hagrid continued to do magic using his pink umbrella, which most likely contained the broken fragments of his wand, or perhaps even his entire wand intact, reconstructed for him by Dumbledore using the Elder Wand.