There seems to be a difference of opinion here. If the subject is not under arrest ( as defined by law/gov.uk) for any offence and has been detained correctly and appropriately under sect 136 - then isn’t 136 (C) more applicable then Sect 32 search - at any location. Police National Legal Database and other Police Forces determine that this is the case.
To help others find the legislation, here are links: s136C MHA 1983 and s32 PACE 1984.
I don’t fully understand your question, and am not sure what you mean by “more applicable”, but did a quick internet search. Essex Police’s document “E0501 Procedure - Section 136 Mental Health Act 1983” (v12, 13/11/20) covers searches under the heading “3.6 Powers to Search”. It describes the purpose (and therefore limitations) of the MHA power:
The new search power is Section 136C which enables a Police Officer to search a person detained under S135 or S136 if the officer has reasonable grounds to believe that the person detained may be a danger to themselves or others and is concealing something which could be used to physically injure themselves or others.
And it concludes by considering how the various search powers interrelate:
Section 136C does not alter the applicability of other existing powers, including powers under section 32 or 54 of PACE, and the general powers of health professionals to search patients in hospitals. The applicability of other such powers does not prevent the new search powers from applying.
Hugely delayed response by me here, but only just noticed that Jonathan tagged me! I’m not great with IT forums, so sorry!
Section 136C doesn’t apply to detention under s136(1) - only to detention under s136(2) or s136(4). Search after “arrest” under s32 PACE always was and still is the power to use after section 136(1) detention because it applies to all “arrests”, not just to arrests for an offence. Detention under s136 is an arrest, in law, even though it’s not an arrest for an offence - see s26 and schedule 2 of PACE for confirmation of that.
The 136C section was introduced specifically to address two problems: powers of search AFTER arrival at a Place of Safety under s136 and all powers of search for warrants under s135(1) or (2). Search after initial s136 detention didn’t need changing, because s32 always covered it.
Hope that helps. 
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