I am aware of the rules surrounding long rests and hit dice. What I am interested in here is whether there is any way apart from long resting to regain any number of hit dice?
Aside from "do what you want" abilities like Wish-ing to recover your hit dice (in whatever in-character phrasing you desire), the answer is No. This appears to be an intentional design decision, such that long rests are the limiting factor on healing. To achieve this, in general, all sources of healing either:
- Use spell slots (only recovered in unbounded sense by long rests for all but warlocks), or
- Spend hit dice (also recovered by long rests)
- Have a recharge cycle directly tied to long rests (e.g. Paladin's Lay on Hands, Celestial Warlock's Healing Light) or, for magic items which generally aren't phrased in terms of long rests, the day cycle (which isn't all that different from per-long-rest in most cases)
- Expend finite resources (e.g. money/magic items) and therefore run out eventually (not technically long rest bounded, but bounded by money, which the game assumes is fairly finite).
In practice, stuff like the Coffeelock exists, and certain class & feat combinations could use exploits like that to produce a meaningful amount of spell-based healing limited only by short rests, not long rests, but hit dice, to my knowledge, have no exploits that allow you to recover them without a long rest.
Variant Rules and Homebrew
As Shadow Ranger's excellent answer states, the simply answer is no. This is likely an intentional design decision to limit the availability of healing without a long rest. However, depending on your game and if your table is open to it there can be other options.
Optional Rule: Healing Surges
Chapter 9 of the DMG contains a number of variant / optional rules that might suit your group. One of these options, Healing Surges, changes how Hit Dice are regained:
This optional rule allows characters to heal up in the thick of combat and works well for parties that feature few or no characters with healing magic, or for campaigns in which magical healing is rare. [...]
Under this optional rule, a character regains all spent Hit Dice at the end of a long rest. With a short rest, a character regains Hit Dice equal to his or her level divided by four (minimum of one die).
Homebrew Rule: Comfy Beds
In my current campaign we are utilising another of the variant rules, Gritty Realism:
This variant uses a short reset of 8 hours and a long rest of 7 days.
Using this rules massively increases the importance of Hit Dice and short rest healing. Particularly in my exploration based campaign where places to safely rest for 7 days are few and far between. Therefore to help the party manage healing and reduce pressure on the Cleric's spell slots I introduced the following homebrew rule:
When completing a short rest, if a character is sleeping in a comfortable bed (such as may be purchased in a nice tavern or home rather than rough camp bed) they regain one hit dice. This hit dice may be spent to recover hit points during the same rest as it was earned.
This house rule encourages the party to engage with the environment, paying for inn stays and making allies rather than camping outside of towns. So far the feedback from my players has been highly positive, particularly from the Cleric (frees up spell slots for more interesting things) and the fighter (party tank needs the most healing).