Make a grid column span the entire row

Here are two interesting sections in the CSS Grid specification:

7.1. The Explicit Grid

Numeric indexes in the grid-placement properties count from the edges
of the explicit grid. Positive indexes count from the start side,
while negative indexes count from the end side.

also here…

8.3. Line-based Placement: the grid-row-start, grid-column-start, grid-row-end, and grid-column-end properties

If a negative integer is given, it instead counts in reverse, starting
from the end edge of the explicit grid.

In other words, when dealing with an explicit grid, which means a grid defined by these properties:

• grid-template-rows
• grid-template-columns
• grid-template-areas
• grid (which is the shorthand for the three properties above, among others)

… you can make a grid area span all columns by setting this rule:

grid-column: 1 / -1;

That tells the grid area to span from the first column line to the last column line, which I believe meets your stated objective:

“We would need to apply something like grid-column: span ALL (if something like that exists), with meaning that ALL = till the end of current row.”

jsFiddle demo

.grid {
  display: grid;
  grid-template-columns: repeat(auto-fill, minmax(100px, 1fr));
  background-color: silver;
}

.grid-second {
  grid-column: 2 / -1;
  background-color: red;
}


/* Not important fancy styles */

.grid div {
  height: 40px;
  text-align: center;
  padding-top: 20px;
}

.grid-another {
  background-color: purple;
  border: 1px solid gray;
}

<div class="grid">
  <div class="grid-first">First</div>
  <div class="grid-second">Second (Want till end)</div>
</div>
<!-- Another same grid -->
<div class="grid">
  <div class="grid-another">1</div>
  <div class="grid-another">2</div>
  <div class="grid-another">3</div>
  <div class="grid-another">4</div>
  <div class="grid-another">1</div>
  <div class="grid-another">2</div>
  <div class="grid-another">3</div>
  <div class="grid-another">4</div>
  <div class="grid-another">1</div>
  <div class="grid-another">2</div>
  <div class="grid-another">3</div>
  <div class="grid-another">4</div>
</div>