In my view, Chris McDowall's Into the Odd is the spiritual successor to Molday's Basic D&D.
As a set of rules, Into the Odd has a laser-fine focus on exploration. The aim of the character creation section of the book is to get you a character quickly and efficiently, so you can begin playing immediately. The rules give you just enough mechanics to adjudicate any risky action that might be needful while exploring, and no more. Space--whether given over to text or illustration-- is treated like the rarest of treasure in the book itself; if it isn't essential, it got left on the cutting room floor.
Shockingly, Into the Odd actually manages to improve on Basic D&D's framework in places. Character generation couldn't be simpler: roll 3d6 for Strength, Dexterity, and Willpower, swap two of those scores if you want, roll 1d6 for Hit Points, cross-reference your Hit Points with your highest ability score on a table to see what you have for starting equipment. Six ability scores? Nah, why waste time dithering about the metaphysical difference between Intelligence and Wisdom. Picking a character class? Nah, you're all adventurers, why pretend otherwise. Shop for gear? Get out there and seek death or glory already.
The resolution mechanics are similarly streamlined: you roll a d20 and hope to equal or roll under one of your ability scores when you take a risky action. Combat works a little differently: if you want to hit something, you hit it automatically; roll damage (modified by any armor the target may have) and subtract what remains from their Hit Points. When all the Hit Points are gone, subtract damage from your Strength score and hope you don't take critical damage or run out of Strength and die.
(If auto-hitting bothers you, here's an add-on from yours truly: each attack is a Strength save. Succeed on the save and proceed to rolling damage; fail and miss.)
The player-facing rules all fit on a single small page. All the information on character advancement fit on a single page. All the principles of running the game fit on one page. The book also comes with a sampler of monsters, treasure, traps, some terse setting information, a sample dungeon to explore, a small hexcrawl to explore, And an "Oddpendium" of random tables. There are descriptions of the various Arcanum--weird technological/magical items that make exploring forsaken areas worth the obvious danger. The only thing that feels like a throwaway is the page on enterprises and detachments, which feels vague and poorly-defined. The art is okay, but nothing really stands out there. Keep in mind, though, that the book is tiny. It does one thing, and does it exceedingly well. The pdf is available here; the third printing of the hardcopy edition is currently available for pre-order.
If I were going to run a fast-and-loose game of dungeon exploration, Into the Odd would be my first choice.