The Giant Toad's Swallow action says:
If the toad dies, a swallowed creature is no longer restrained by it and can escape from the corpse using 5 feet of movement, exiting prone.
Is there any way to get out without killing the Giant Toad first?
There are other ways small creatures can escape, aside from trying to kill the Giant Frog. Here is an incomplete list.
Find Familiar's vision swap (Action) + Misty Step (Bonus Action)
Clairvoyance (Round 1 Action) + Misty Step (Round 2 Bonus Action)
Enlarge/Reduce or Wildshape: The small creature can Enlarge itself or Reduce the Giant Frog. Either way, the Frog may have to spit out the swallowed creature. This is not necessarily what happens, depending on the DM. But as the Giant Frog can only swallow small creatures in the first place, it makes sense that if that swallowed target grew in size while swallowed, the Frog has to spit it out.
High level escape spells: Blink, Freedom of Movement, Dimension Door, Teleportation, Plane Shift, Etherealness
Gaseous Form: the creature can pass through pin-hole sized openings while in this form.
Stinking Cloud: on a failed Con save, a target inside the area spends its turn "retching and reeling" -- as in, vomiting.
Pushing Attack: On a hit, the Giant Frog must succeed a Strength save or be pushed back 15 ft away from you. The exact thing that happens here is DM-dependent, though, as it's easy to imagine a DM who will disallow this on the basis of reality (you can't push someone away from you if you are inside them, after all).
Generally, enchantment-based spells will not work because you need to be able to see the target. However, you are blinded while inside this monster and thus can't see.
I looked over the Monster Manual and the MM Errata, and it looks like the answer is no. Per the RAW, the only way out of a Giant Frog or Giant Toad is to kill them.
Other creatures in the MM such as the Behir, Kraken, Purple Worm, Remorhaz, and Tarrasque all have rulings that if they take beyond a certain threshold of damage from a creature inside of them, they must make a Con save or regurgitate whatever they have swallowed. Giant Frogs and Toads are the only creatures in the Monster Manual that do not appear to have this limitation.
This may be an oversight, but it has never been addressed by WotC. It may simply be that Giant Frogs and Giant Toads have little enough HP that, if you hit them hard enough to make them throw you up....you killed them. This sort of makes sense, because the lowest damage threshold to trigger a 'Save or regurgitate' effect is 30 points of damage. And a Giant Toad only has 39...
Yes, you have several solutions to escape from inside the toad or to be extracted from outside with the help of other party members.
For getting out of the Toad, the simplest way would be to teleport . (You don't need to see where you want to go)
Other solutions (subject to DM approval)
After that, I the other options I see depend on your DM. You could try to force the Toad to vomit by creating a reaction in her stomach (by creating gas or fire inside it [or disgusting material that would provoke the toad to gag])
Another way would be to dominate the Toad to spit the swallowed character (for example with dominate beast). And if the DM argues that you cannot force the toad to spit the character, you could force it to open its mouth as wide as possible.
A low level solution would be suggestion.
If you give the condition paralyzed or unconscious to the Toad, you could open its mouth and try to extract the character (you could to that using a wand of paralysis). As noted by @Mindwin, this solution could be subject to the DM ruling.