今朝テレビを見てきた = I watched a tv show this morning (, then I came) .
    いままでたくさんのテレビを見てきた = I have watched many tv shows until now.
    彼に電話をかけてくる = I'll call him (then come back).
    彼が電話をかけてくる = He'll call me.
    寒いから明日はコートを着てくる = It's cold, I'll come with a coat tomorrow.
    ちょと待って、コートを着てくる = Hold on a sec, I'm going to put my coat on (then come back).
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1. Spacial movement - sequential
Do something, then come. Works with most verbs describing actions.
ビールを買ってくる I'll go buy some beer, then come back.
ご飯を食べてきた I ate my meal, then came here.​

2. Spacial movement - possession
Come with/without something. Works with verbs meaning carry, possess, etc.
犬を連れてくる I'll come with a dog (= I'll come bringing a dog)
荷物を運んできた I brought the luggage (= I came carrying the luggage).
Your 置いてきちゃった example in another thread can be classified as one of these:
I failed to bring it = I came leaving it behind.​

3. Spacial movement - method
With verbs involving change of location such as run, walk, swim, fly.
走ってくる = I'll come running.
飛んできた = I came flying.​

4. Spacial movement - attire
With verbs meaning to put something on, to wear something.
帽子をかぶってくる = I'll come wearing a hat.
コートを着てきた = I came wearing a coat.​

5. Spacial movement - movement of the subject
With verbs related to destination/direction, such as enter, exit, rise, sink, leave home, come home, etc.
家に帰ってくる = I'll come home (when the speaker is at home)
太陽が登ってきた = The sun came up (from the perspective of the speaker).​

6. Spacial movement - movement of the object
With verbs such as send, bring, call (telephone)
友達が電話をかけてくる = My friend will call me.
母が手紙を送ってきた = My mother sent me a letter.​

7. Time course - continuation
When the action/state has continued (V-てきた) or will continue (V-ていく).
To state about the matters up to the present time, past tense V-てきた is used.
To state about the matters after the present time, present/future tense V-ていく is used.
これまで幸せに生きてきた = I've had a happy life so far.
これから幸せに生きていく = I'll have a happy life from now on.
​
8. Transition - state
With verbs such as become, change, etc. Describes the change of state.
これから寒くなってくる = It's going to grow colder (It is not cold yet).
最近太ってきた = I became fat these days (I was not fat earlier).​

9. Transition - sense
With sense verbs such as hear, see, smell. Describes that the stimulus reaches the speaker.
角を曲がると家が見えてくる = When I turn the corner, I'll see my house. (The house will come into view)
いい匂いがしてきた = I smell something good (The aroma has reached me.)​

For the record, these are my interpretation of the information on the following pages:
「～てくる・～ていく」表現（１） | 日本語教師の広場
「～てくる、～ていく」表現（２） | 日本語教師の広場
「～てくる、～ていく」表現（３） | 日本語教師の広場
「～てくる、～ていく」表現（４） | 日本語教師の広場
「～てくる、～ていく」表現（５） | 日本語教師の広場