Why wouldn't an ASDIC work immediately after a depth charge attack?

The quote may be referring to the physical systems implementing ASDIC. Shock and vibration shielding are major issues in naval architecture. Insufficiently shock shielded systems may have required maintenance after being shocked, for example by being near a recently executed depth charge attack. At least some British commonwealth WWII vessels lacked full shock protection on their ASDIC systems from the shocks induced by their own depth charge attacks.

Anti-submarine warfare division, Navy Office, Melbourne (1943-07) “SOUTH-WEST PACIFIC ANTI-SUBMARINE REPORT JULY, 1943” ACB0233/43(2) [ http://www.navy.gov.au/sites/default/files/documents/1943_July.pdf ] p. 9:

"KALGOORLIE" made an attack on a possible contact at 1808, and two minutes later "WARRNAMBOOL" had an echo on the bear­ing reported by "KALGOORLIE", range 700 yards. Four charges were dropped, the starboard thrower again misfiring. The concussion of the last charge put the Asdic set out of action. One valve had been jarred from its socket and the heterodyne condenser had moved from setting "E" to setting "B".

To release depth charge ship would have to go below sonar's minimum range

Sonars (or ASDIC in British terminology) were relatively simple in WW2. Sound would be sent in one direction, it would propagate trough the water, and would bounce back from any underwater object in its path (submarine for example). Knowing the speed of sound in the water it was possible to roughly calculate range and bearing of that object. But if the object gets to close to the sonar, it could not detect it. The article about Hedgehog you posted explains that :

The system was developed to solve the problem of the target submarine disappearing from the attacking ship's ASDIC when the ship came within the sonar's minimum range. Due to the speed of sound in water, the time taken for the 'ping' echo to return to the attacking ship from the target submarine became too short to allow the human operator to distinguish the returning audible echo from that of the initial sound pulse emitted by the sonar – the so-called "instantaneous echo", where the output sound pulse and returning echo merge. This "blind spot" allowed the submarine to make evasive manoeuvres undetected while the ship was out of range for depth charge attack. Hence, the submarine was effectively invisible to the sonar as the ship came within the sonar's minimum range. The solution was a weapon mounted on the foredeck that discharged the projectiles up and over that carrying ship's bow, to land in the water some distance in front of the ship while the submarine was still outside the sonar's minimum range.

Depth charges on the other hand were more cumbersome weapons. As you can see from the link, they were usually dropped in the wake of the ship. This means that attacking ship had to cross over submarine, i.e. loose contact for some time. Unlike hedgehogs, depth charges would explode every time (at predetermined depth). Attacking ship would lay pattern of them, wait till they explode, get out of the minimum range and then try to reacquire target providing the cavitation has settled down.

Providing all necessary steps, the fact that sonars rarely worked at speeds above 15 kts and range under 300 yards, plus limitations of sending pings at intervals that lasted around 5 seconds in 5 degree arc (to allow sound to return back) , it could really last up to 15 minutes before you could start search again if first depth charge attack was unsuccessful.

I think that the key to this is considering the reflected waves and their interactions with the environment and each other.

The initial explosion will send out shockwaves in all directions. These shockwaves will bounce off any surface, especially the sea floor and the surface with the air. Importantly, these reflected waves will also reflect as they hit any surface (for example, the initial reflection from the seabed will bounce up and reflect from the surface) and those reflections will do the same. As these surfaces are generally going to be irregular so the reflected shockwaves will very quickly become a jumble of noise.

It's also worth considering that depth charges usually dropped in patterns, so there wouldn't be just a single explosion but a series of them. The combination of shockwaves and reflections (and interference between the shockwaves and reflections) will make things very noisy indeed.

The initial explosions are orders of magnitude stronger (i.e. louder) than the ASDIC sound waves, and although each reflection loses energy, it takes time for the reflections to lose sufficient energy that they become insignificant in comparison to the ASDIC signals.

An underwater explosion creates turbulence in the water, creates bubbles, and perhaps mixes waters of varying temperatures or salinities, all of which affect the refraction of sound in water. For ASDIC to work best the water should be h*m*geneous, with laminar flow only.