You can use CommandStats to simulate an OR gate while reducing the amount of required command blocks (since you'd essentially be running two commands for the price of one).
Prerequisites
Objective to hold the CommandStats result.
/scoreboard objectives add Condition dummy
Single armor stand to summon, which could also be used for other chains that needs it (as its score is to be reset at the beginning of the chains). This will be the entity subject to CommandStats.
/summon ArmorStand ~ ~ ~ {Tags:["conditional"]}
Mechanism
Example mechanism (you can swap out the impulse for a repeating block):

Command blocks 2 and 3 will be the initial conditional blocks, where if either are successful you would run more commands. Place a temporary command block above each of them, place the following command inside it, and activate it (you would otherwise just run the command yourself, but the character limit was exceeded):
/stats block ~ ~-1 ~ set SuccessCount @e[type=ArmorStand,tag=conditional,score_Condition=0] Condition
Now if either of those two commands run, the armor stand will have their "Condition" score set equal to the number of successful iterations of the command.
However, it will only target the armor stand if it has a "Condition" score of 0. This means that if either of them are successful, the score will be 1 and will remain at 1 even if the next command block is unsuccessful.
Commands
The following are the commands for the example mechanism above.
Set the armor stand's "Condition" score to 0. This is needed so it can be targeted by this chain again.
/scoreboard players set @e[type=ArmorStand,tag=conditional] Condition 0
One of the conditionals to check. The success of this command will trigger its stored CommandStats.
/scoreboard players test #DAYTIME daytime 0 1000
The second conditional to check.
/scoreboard players test #DAYTIME daytime 12000 13000
After the conditionals have finished, the armor stand will have a score of 1 if either of them were successful. You can then detect this armor stand with a score of 1.
/testfor @e[type=ArmorStand,tag=conditional,score_Condition_min=1,c=1]
Command(s) to run if either of the conditionals were successful.
/say The time is either 0-1000 or 12000-13000.