A day without midnight

There are several. As of 2020-04, there are 14 such time zones in 10 countries: Paraguay, Cuba, Chile, Greenland/Denmark, Jordan, Lebanon, Syria, Palestine, Iran, and Azores/Portugal.

Before 2019-04, the list included Brazil using the America/Sao_Paulo time zone.

$ perl -MDateTime -E'say DateTime->new(
     year => 2013, month => 10, day => 20, hour => 12,
     time_zone => "America/Sao_Paulo")->truncate( to => "day" )->ymd;'
Invalid local time for date in time zone: America/Sao_Paulo

You can get around the problem by switching to the "floating" tome zone before getting the date:

$ perl -MDateTime -E'say DateTime->new(
     year => 2013, month => 10, day => 20, hour => 12,
     time_zone => "America/Sao_Paulo")
   ->set_time_zone("floating")
   ->truncate( to => "day" )
   ->ymd;'
2013-10-20

See this documentation.

I've thrown together a small program in Java to find all zones that contain a date in 2020 where the start of the date is not midnight. Its output lists all the time zones on my machine and version of Java matching your requirements as of 2020, as well as the date & time of the non-midnight start of day.

import java.time.*;
import java.time.zone.ZoneRulesProvider;
import java.util.stream.Stream;

public class ListTimeZonesWithoutMidnight {
    public static void main(String[] args) {
        System.out.printf("Java vendor & version: %s %s\n", System.getProperty("java.vendor"), Runtime.version());
        System.out.printf("Zone rules versions: %s\n", ZoneRulesProvider.getVersions("UTC").keySet());
        System.out.println();

        ZoneId.getAvailableZoneIds().stream().sorted().map(ZoneId::of)
                .flatMap(zoneId -> getNonMidnightStartOfDays(Year.of(2020), zoneId))
                .forEachOrdered(System.out::println);
    }

    private static Stream<LocalDate> getAllDates(Year year) {
        return year.atDay(1).datesUntil(year.plusYears(1).atDay(1));
    }

    private static Stream<ZonedDateTime> getNonMidnightStartOfDays(Year year, ZoneId zoneId) {
        return getAllDates(year).map(d -> d.atStartOfDay(zoneId))
                .filter(d -> !d.toLocalTime().equals(LocalTime.MIDNIGHT));
    }
}

Output:

Java vendor & version: AdoptOpenJDK 14.0.1+7
Zone rules versions: [2019c]

2020-10-04T01:00-03:00[America/Asuncion]
2020-03-08T01:00-04:00[America/Havana]
2020-09-06T01:00-03:00[America/Santiago]
2020-03-29T01:00Z[America/Scoresbysund]
2020-03-27T01:00+03:00[Asia/Amman]
2020-03-29T01:00+03:00[Asia/Beirut]
2020-03-27T01:00+03:00[Asia/Damascus]
2020-03-27T01:00+03:00[Asia/Gaza]
2020-03-27T01:00+03:00[Asia/Hebron]
2020-03-21T01:00+04:30[Asia/Tehran]
2020-03-29T01:00Z[Atlantic/Azores]
2020-09-06T01:00-03:00[Chile/Continental]
2020-03-08T01:00-04:00[Cuba]
2020-03-21T01:00+04:30[Iran]

Those 14 zone names are in 10 countries: Paraguay, Cuba, Chile, Greenland/Denmark, Jordan, Lebanon, Syria, Palestine, Iran, and Azores/Portugal.