You have to change the value of textdistance.

\setuphead
   [subsubsection]
   [alternative=text,
    distance=0pt,
    textdistance=.5em]

\starttext

\startchapter[title={First}]

\startsubsubsection
\input ward
\stopsubsubsection

\stopchapter

\stoptext

Wolfgang
> Idris Samawi Hamid ادريس سماوي حامد <mailto:Idris.Hamid@colostate.edu>
> 25. Juni 2018 um 04:45
> Dear gang,
>
> Here's an interesting one, attached and below:
>
> =======
> \setuphead[chapter]   [alternative=normal]
> \setuphead[subsection][alternative=text]
> \setuphead[subsection][criterium=local,distance=0em,sectionstopper=.]
>
> \starttext
>
> \startchapter[title={First},reference={}]
> \startsubsection%[title={First}]
> \input ward
> \stopsubsection
> \startsubsection%[title={Second}]
> \input ward
> \stopsubsection
> \stopchapter
>
> \startchapter[title={Second},reference={}]
> \startsubsection%[title={}]
> \input ward
> \stopsubsection
> \startsubsection%[title={}]
> \input ward
> \stopsubsection
> \stopchapter
> \stoptext
> =======
>
> Our subsections do not have titles. The space between the subsection 
> numeral and the text is too large, even with
>
> distance=0em
>
> Negative distances don't behave well - toggle lines 4 and 5.
>
> What can we do to control the space between numeral and text? Thank 
> you in advance.
>
> Best wishes
> Idris
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> ___________________________________________________________________________________
> Wolfgang Schuster <mailto:schuster.wolfgang@gmail.com>
> 25. Juni 2018 um 13:41
> To number paragraphs you can create a counter with \definelabel.
>
>
> \definelabel[Paragraph][text=,numberstopper={.},headcommand=\groupedcommand{}{ }]
>
> \starttext
>
> \startchapter[title={First},reference={}]
>
> \Paragraph\input ward
>
> \Paragraph\input ward
>
> \stopchapter
>
> \startchapter[title={Second},reference={}]
>
> \Paragraph\input ward
>
> \Paragraph\input ward
>
> \stopchapter
>
> \stoptext
>
>
> Wolfgang
>
>
> Idris Samawi Hamid ادريس سماوي حامد <mailto:Idris.Hamid@colostate.edu>
> 25. Juni 2018 um 14:58
> On Mon, 25 Jun 2018 05:41:15 -0600, Wolfgang Schuster 
> <schuster.wolfgang@gmail.com> wrote:
>
>> To number paragraphs you can create a counter with \definelabel.
>
> Yes - and tested some paragraph options yesterday. Many thanks for the 
> sample below.
>
> But this particular project is very large with countless 
> cross-references already in place that depend on the subsectioning 
> mechanism. Is there a workaround that will avoid major surgery to the 
> main text?
>
> Also, subsubsections and subsubsubsections are still needed, as well 
> as resetting the counters after each chapter etc..
>
> In other words, what's needed is all the features of subsection 
> without the titling. Just as
>
> \start|stoptitle \start|stopsubject [etc.]
>
> give titling without sectioning numerals, we need something like
>
> \start|stoplabeling \start|stopsublabeling
>
> to get sectioning numerals without titling.
>
> (This is usually not a big deal except in cases like [alternative=text])
>
> (And on a very tight deadline so there is little (‍= no) time to 
> rebuild structure from scratch. Better to configure subsection to do 
> what's needed, and perhaps use a different method for the next big 
> project.)
>
> So is there a way this can be done? Many thanks for your insights and 
> help.
>
> Idris
>
>> \definelabel[Paragraph][text=,numberstopper={.},headcommand=\groupedcommand{}{ 
>> }]
>>
>> \starttext
>>
>> \startchapter[title={First},reference={}]
>>
>> \Paragraph\input ward
>>
>> \Paragraph\input ward
>>
>> \stopchapter
>>
>> \startchapter[title={Second},reference={}]
>>
>> \Paragraph\input ward
>>
>> \Paragraph\input ward
>>
>> \stopchapter
>>
>> \stoptext
>
> Can this be made into
>
>> Wolfgang
>>
>>> Idris Samawi Hamid ادريس سماوي حامد <mailto:Idris.Hamid@colostate.edu>
>>> 25. Juni 2018 um 04:45
>>> Dear gang,
>>>
>>> Here's an interesting one, attached and below:
>>>
>>> =======
>>> \setuphead[chapter]   [alternative=normal]
>>> \setuphead[subsection][alternative=text]
>>> \setuphead[subsection][criterium=local,distance=0em,sectionstopper=.]
>>>
>>> \starttext
>>>
>>> \startchapter[title={First},reference={}]
>>> \startsubsection%[title={First}]
>>> \input ward
>>> \stopsubsection
>>> \startsubsection%[title={Second}]
>>> \input ward
>>> \stopsubsection
>>> \stopchapter
>>>
>>> \startchapter[title={Second},reference={}]
>>> \startsubsection%[title={}]
>>> \input ward
>>> \stopsubsection
>>> \startsubsection%[title={}]
>>> \input ward
>>> \stopsubsection
>>> \stopchapter
>>> \stoptext
>>> =======
>>>
>>> Our subsections do not have titles. The space between the subsection
>>> numeral and the text is too large, even with
>>>
>>> distance=0em
>>>
>>> Negative distances don't behave well - toggle lines 4 and 5.
>>>
>>> What can we do to control the space between numeral and text? Thank
>>> you in advance.
>