Oracle 18c

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with tab as(
select 'date1' as dat,  '09:00' as  start_hour, '09:30' as end_hour from dual union all
select 'date1' as dat,  '10:30' as  start_hour, '11:30' as end_hour from dual union all
select 'date1' as dat,  '13:00' as  start_hour, '15:00' as end_hour from dual 
)
SELECT COUNT(*)
  FROM   tab
  WHERE  start_hour <= '09:10' --:new_end_hour
  AND    end_hour   >= '07:00' --:new_start_hour
  AND    dat = 'date1'
  ;


COUNT(*)


1


with tab as(
select 'date1' as dat,  '09:00' as  start_hour, '09:30' as end_hour from dual union all
select 'date1' as dat,  '10:30' as  start_hour, '11:30' as end_hour from dual union all
select 'date1' as dat,  '13:00' as  start_hour, '15:00' as end_hour from dual 
)
SELECT COUNT(*)
  FROM   tab
  WHERE  start_hour <= '08:10' --:new_end_hour
  AND    end_hour   >= '07:00' --:new_start_hour
  AND    dat = 'date1'
  ;


COUNT(*)


0


with tab as(
select 'date1' as dat,  '09:00' as  start_hour, '09:30' as end_hour from dual union all
select 'date1' as dat,  '10:30' as  start_hour, '11:30' as end_hour from dual union all
select 'date1' as dat,  '13:00' as  start_hour, '15:00' as end_hour from dual 
)
SELECT COUNT(*)
  FROM   tab
  WHERE  ('09:00' between start_hour and end_hour
  or    '09:10' between start_hour and end_hour
  )
  AND    dat = 'date1'
  ;


COUNT(*)


1


with tab as(
select 'date1' as dat,  '09:00' as  start_hour, '09:30' as end_hour from dual union all
select 'date1' as dat,  '10:30' as  start_hour, '11:30' as end_hour from dual union all
select 'date1' as dat,  '13:00' as  start_hour, '15:00' as end_hour from dual 
)
SELECT COUNT(*)
  FROM   tab
  WHERE  ('08:00' between start_hour and end_hour
  or    '09:10' between start_hour and end_hour
  )
  AND    dat = 'date1'
  ;


COUNT(*)


1


with tab as(
select 'date1' as dat,  '09:00' as  start_hour, '09:30' as end_hour from dual union all
select 'date1' as dat,  '10:30' as  start_hour, '11:30' as end_hour from dual union all
select 'date1' as dat,  '13:00' as  start_hour, '15:00' as end_hour from dual 
)
SELECT COUNT(*)
  FROM   tab
  WHERE  ('08:00' between start_hour and end_hour
  or    '08:10' between start_hour and end_hour
  )
  AND    dat = 'date1'
  ;


COUNT(*)


0